Zariski topology

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###Context###
#### Topology
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[[!include topology - contents]]
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#### Algebra
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[[!include higher algebra - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The _Zariski topology_ is a [[topological space|topology]] on the [[prime spectrum of a commutative ring]]. It serves as the basis for much of [[algebraic geometry]].

{#Outline} We consider the definition in increasing generality and sophistication:

1. First we discuss the naive Zariski topology [on affine spaces](#OnAffineSpace) $k^n$, consider the classical proofs and discover thereby the special role of prime ideals and maximal ideals;

1. then we turn to the modern definition of the Zariski topology [on affine varieties](#OnAffineVarieties) $Spec(R)$ which takes the concept of prime (and maximal) ideals as primary, and again we provide the classical arguments;

1. finally we discuss the abstract [[category theory|category theoretic]] perspective on these matters [in terms of Galois connections](#InTermsOfGaloisConnections) and obtain slick category theoretic proofs of all the previous statements.

Starting with [[affine space]] $k^n$, then the idea of the Zariski topology is to take as the _[[closed subsets]]_ those defined by the vanishing of any set of [[polynomials]] over $k$ in $n$ [[variables]], hence the solution sets to [[equations]] of the form

$$
  \underset{i \in I}{\forall}
  \left(
     f_i(x_1, \cdots, x_n) = 0
  \right)
$$

for $f_i \in k[X_1, \cdots, X_n]$ [[polynomials]]. The [[open subsets]] of the topology are the [[complements]] of these _vanishing sets_.

It is clear that the vanishing set such a set of polynomials depends only on the [[ideal]] in the [[polynomial ring]] which is generated by them. Under this translation then forming the intersection of closed subsets corresponds to forming the sum of these ideals, and forming the union of closed subsets corresponds to forming the product of the corresponding ideals. This way the Zariski topology establishes a dictionary between topological concepts of the affine space $k^n$, and algebra inside the polynomial ring.

In particular one finds that the [[irreducible closed subsets]] of the Zariski topology correspond to the [[prime ideals]] in the polynomial ring (prop. \ref{PrimeVanishingIdealOfIrreducibleZariskiClosed} and prop. \ref{PrimeIdealClosedsubspaceBijection} below), and that the [[closed points]] correspond to the [[maximal ideals]] among these (prop. \ref{MaximalIdealsAreClosedPoints}).

This motivates the modern refinement of the concept of the Zariski topology, where one considers any [[commutative ring]] $R$ and equips its set of [[prime ideals]] with a topology, by direct analogy with the previously naive affine space $k^n$, which is recovered with $R$ a polynomial ring and restricting attention to the maximal ideals (example \ref{AffinSpaceAsPrimeSpectrum} below).

These sets of prime ideals of a ring $R$ equipped with the Zariski topology are called the (topological spaces underlying) the _[[prime spectrum of a commutative ring]]_, denoted $Spec(R)$.

The Zariski topology is in general not [[Hausdorff topological space|Hausdorff]] (example \ref{AffineSpaceOverInfiniteFieldNotHausdorff} below) which makes it sometimes be regarded as an "exotic" type of topology. But it is in fact [[sober topological space|sober]] (prop. \ref{ZariskiTopologyIsSober} below) and hence as well-behaved in this respect as general [[locales]] are.

## On affine space
 {#OnAffineSpace}

We consider here, for $k$ a [[field]], the [[vector space]] $k^n$ equipped with a Zariski topology. This is the original definition of Zariski topology, and serves well to motivate the concept, but eventually it was superceded by a more refined concept of Zariski topologies of prime spectra, discussed in the next subsection [below](#OnAffineVarieties). In example \ref{AffinSpaceAsPrimeSpectrum} below we reconsider the naive case of interest in this subsection here from that more refined perspective.

### Definition

+-- {: .num_defn  #ZariskiOpenSubsetsOnAffineSpace}
###### Definition
**(Zariski topology on affine space)**

Let $k$ be a [[field]], let $n \in \mathbb{N}$, and write $k[X_1, \cdots, X_n]$ for the [[set]] of [[polynomials]] in $n$ [[variables]] over $k$.

For $\mathcal{F} \subset k[X_1, \cdots, X_n]$ a subset of polynomials, let the  subset $V(\mathcal{F}) \subset k^n$ of the $n$-fold [[Cartesian product]] of the underlying set of $k$ (the _vanishing set_ of $\mathcal{F}$) be the subset of points on which all these polynomials jointly vanish:

$$
  V(\mathcal{F})
    \coloneqq
  \left\{
     (a_1, \cdots, a_n) \in k^n
     \,\vert\,
     \underset{f \in \mathcal{F}}{\forall} f(a_1, \cdots, a_n) = 0
  \right\}
  \,.
$$

These subsets are called the _Zariski [[closed subsets]]_.

Write

$$
  \tau_{\mathbb{A}^n_k}
  \;\coloneqq\;
  \left\{
    k^n \backslash V(\mathcal{F}) \subset k^n
    \,\vert\,
    \mathcal{F} \subset k[X_1, \cdots, X_n]
  \right\}
$$

for the set of [[complements]] of the Zariski closed subsets. These are called the _Zariski [[open subsets]]_ of $k^n$.

=--

+-- {: .num_prop #VerifyingZariskiTopologyOnAffineSpace}
###### Proposition
**(Zariski topology is well defined)**

Assuming [[excluded middle]], then:

For $k$ a [[field]] and $n \in \mathbb{N}$, then the Zariski open subsets of $k^n$ (def. \ref{ZariskiOpenSubsetsOnAffineSpace}) form a [[topological space|topology]]. The resulting [[topological space]]

$$
  \mathbb{A}^n_k
  \;\coloneqq\;
  \left(
     k^n, \tau_{\mathbb{A}^n_k}
  \right)
$$

is also called the $n$-dimensional _[[affine space]]_ over $k$.

=--

+-- {: .proof}
###### Proof

We need to show for $\{\mathcal{F}_i \subset k[X_1, \cdots, X_n]\}_{i \in I}$ a set of subsets of polynomials that

1.  $\underset{i \in I}{\cup} \left(k^n \backslash V(\mathcal{F}_i)\right) = k^n \backslash V(\mathcal{F}_\cup)$ for some $\mathcal{F}_\cup \subset k[X_1, \cdots, X_n]$;

1. if $I$ is [[finite set|finite]] then $\underset{i \in I}{\cap} \left( k^n \backslash V(\mathcal{F}_{\cap})\right) = k^n \backslash \mathcal{F}_\cap$ for some $\mathcal{F}_{\cap} \subset k[X_1, \cdots, X_n]$.

By [[de Morgan's law]] for [[complements]] (and using [[excluded middle]]) this is equivalent to

1.  $\underset{i \in I}{\cap} V(\mathcal{F}_i) = V(\mathcal{F}_\cup)$ for some $\mathcal{F}_\cup \subset k[X_1, \cdots, X_n]$;

1. if $I$ is [[finite set|finite]] then $\underset{i \in I}{\cup} V(\mathcal{F}_i) =  V(\mathcal{F}_{\cap})$ for some $\mathcal{F}_{\cap} \subset k[X_1, \cdots, X_n]$.

We claim that we may take

1. $\mathcal{F}_\cup = \underset{i \in I}{\cup} \mathcal{F}_i$

1. $\mathcal{F}_{\cap} = \underset{i \in I}{\prod} \mathcal{F}_i \coloneqq \left\{ \underset{i \in I}{\prod} f_i \,\vert\, f_i \in \mathcal{F}_i \right\}$.

(In the second line we have the set of all those polynomials which arise as products of polynomials with one factor from each of the $\mathcal{F}_i$.)

Regarding the first point:

$$
  \begin{aligned}
    & (a_1, \cdots, a_n) \in \underset{i \in I}{\cap} V(\mathcal{F}_i)
    \\
    \Leftrightarrow\;
     &
     \underset{i \in I}{\forall}
     \left(
        (a_1, \cdots, a_n) \in V(\mathcal{F}_i)
     \right)
     \\
     \Leftrightarrow\;
    &
     \underset{i \in I}{\forall}
     \left(
       \underset{f \in \mathcal{F}_i}{\forall}
       \left(
         f(a_1, \cdots, a_n) = 0
       \right)
     \right)
    \\
    \Leftrightarrow\;
    &
    \underset{f \in \underset{i \in I}{\cup} \mathcal{F}_i}{\forall}
    \left(
       f(a_1, \cdots, a_n) = 0
    \right)
    \\
    \Leftrightarrow\;
    &
    (a_1, \cdots, a_n)
    \in
    V\left(
      \underset{i \in I}{\cup} \mathcal{F}_i
    \right)
  \end{aligned}
$$

Regarding the second point, in one direction we have the immediate implication

$$
  \begin{aligned}
    &
    (a_1, \cdots, a_n) \in \underset{i \in I}{\cup} V(\mathcal{F}_i)
    \\
    \Leftrightarrow\;
    &
    \underset{i \in I}{\exists}
      \left(
      \underset{f \in \mathcal{F}_i}{\forall}
      \left(
        f(a_1, \cdots, a_n) = 0
      \right)
    \right)
    \\
    \Rightarrow \;
    &
    \underset{(f_i) \in \underset{i \in I}{\prod} \mathcal{F}_i}{\forall}
    \left(
      \underset{i \in I}{\prod} f_i(a_1, \cdots, a_n) = 0
    \right)
    \\
    \Leftrightarrow\;
    &
    (a_1, \cdots, a_n)
    \in
    V\left(
      \underset{i \in I}{\prod} \mathcal{F}_i
    \right)
    \,.
  \end{aligned}
$$

For the converse direction we need to show that

$$
  \left(
    (a_1 , \cdots , a_n)
    \in
      V\left(
        \underset{i \in I}{\prod} \mathcal{F}_i
      \right)
   \right)
     \;\Rightarrow\;
   \left(
     (a_1, \cdots, a_n)
     \in
     \underset{i \in I}{\cup} V(\mathcal{F}_1)
   \right)
  \,.
$$

hence that

$$
  \left(
    \underset{(f_i) \in \underset{i \in I}{\prod} \mathcal{F}_i }{\forall}
    \left(
      \underset{i \in I}{\prod} f_i(a_1, \cdots, a_n) = 0
    \right)
  \right)
  \;\Rightarrow\;
  \left(
    \underset{i \in I}{\exists}
    \left(
      \underset{f_i \in \mathcal{F}_i}{\forall}
      \left(
         f_i(a_1, \cdots, a_n) = 0
      \right)
    \right)
  \right)
  \,.
$$

By [[excluded middle]], this is equivalent to its [[contraposition]], which by [[de Morgan's law]] is

$$
  \left(
    \underset{i \in I}{\forall}
    \left(
      \underset{f_i \in \mathcal{F}_i}{\exists}
      \left(
         f_i(a_1, \cdots, a_n) \neq 0
      \right)
    \right)
  \right)
  \;\Rightarrow\;
  \left(
    \underset{(f_i) \in \underset{i \in I}{\prod} \mathcal{F}_i }{\exists}
    \left(
      \underset{i \in I}{\prod} f_i(a_1, \cdots, a_n) \neq 0
    \right)
  \right)
  \,.
$$

This now is true by the assumption that $k$ is a [[field]]: If all factors $f_i(a_1, \dots a_n) \in k$ are non-zero, then their product $\underset{i \in I}{\prod} f_i(a_1, \cdots, a_n) \in k$ is non-zero.

=--

### Properties

#### Topological closures

+-- {: .num_prop}
###### Proposition

For $k$ a [[field]] and $n \in \mathbb{N}$, consider a [[subset]]

$$
  S \subset k^n
$$

of the underlying set of the $n$-fold [[Cartesian product]] of $k$ with itself. Then the [[topological closure]] $Cl(S)$ of this subset with respect to the Zariski topology $\tau_{\mathbb{A}^n_k}$ (def. \ref{ZariskiOpenSubsetsOnAffineSpace}) is the vanishing set of all those polynomials that vanish on $S$:

$$
  Cl(S)
  \;=\;
  V \left(
      \left\{
        f \in k[X_1, \cdots, X_n]
        \,\vert\,
        \underset{(a_1, \cdots, a_n) \in S}{\forall} f(a_1, \cdots, a_n) = 0
      \right\}
    \right)
  \,.
$$

=--

+-- {: .proof}
###### Proof

We compute as follows:

$$
  \begin{aligned}
    Cl(S)
     & \coloneqq
    \underset{ {C \subset k^n \, \text{closed}} \atop {C \supset S} }{\cap}
    C
    \\
    & =
    \underset{
       { \mathcal{F} \subset k[X_1, \cdots, X_n] }
       \atop
       { S \subset V(\mathcal{F}) }
    }{\cap} V(\mathcal{F})
    \\
    & =
    V\left(
      \underset{
         { \mathcal{F} \subset k[X_1, \cdots, X_n] }
         \atop
         { S \subset V(\mathcal{F}) }
      }{\cup} \mathcal{F}
      \right)
    \\
    & =
    V \left(
      \left\{
        f \in k[X_1, \cdots, X_n]
        \,\vert\,
        \underset{(a_1, \cdots, a_n) \in S}{\forall} f(a_1, \cdots, a_n) = 0
      \right\}
    \right)
    \,.
  \end{aligned}
$$

Here the first equality is the definition of [[topological closure]], the second is the definition of closed subsets in the Zariski topology (def. \ref{ZariskiOpenSubsetsOnAffineSpace}), the third is the expression of intersections of these in terms of unions of polynomials as in the proof of prop. \ref{VerifyingZariskiTopologyOnAffineSpace}, and then the last one is immediate.

=--

#### Irreducible closed subsets as prime ideals

In every [[topological space]] the [[irreducible closed subsets]] play a special role, as being precisely the points in the space as seen in its incarnation as a [[locale]] ([this prop.](irreducible%20closed%20subspace#FrameHomomorphismsToPointAreIrrClSub)). The following shows that in the Zariski topology the irreducible closed subsets all come from [[prime ideals]] in the corresponding [[polynomial ring]], and that when the [[ground field]] is [[algebraically closed field|algebraically closed]], then they are in fact in bijection to the prime ideals. See also at _[[schemes are sober]]_.

+-- {: .num_defn #VanishingIdeal}
###### Definition
**([[vanishing ideal]] of Zariski closed subset)**#

Let $k$ be a [[field]], and let $n \in \mathbb{N}$. Then for $V(\mathcal{F}) \subset k^n$ a Zariski closed subset, according to def. \ref{ZariskiOpenSubsetsOnAffineSpace}, hence for $\mathcal{F} \subset k[X_1, \cdots, X_n]$ a set of polynomials, write

$$
  I(V(\mathcal{F}))
  \subset
  k[X_1, \cdots, X_n]
$$

for the maximal subset of polynomials that still has the same joint vanishing set:

$$
  I(V(\mathcal{F}))
    \;\coloneqq\;
  \left\{
    f \in k[X_1, \cdots, X_n]
    \,\vert\,
    \underset{(a_1, \cdots, a_n) \in V(\mathcal{F})}{\forall} f(a_1, \cdots, a_n) = 0
  \right\}
  \,.
$$

This set is clearly an [[ideal]] in the [[polynomial]] [[ring]]  $k[X_1, \cdots, X_n]$, called the _[[vanishing ideal]]_ of $V(\mathcal{F})$.

=--

+-- {: .num_prop #PrimeVanishingIdealOfIrreducibleZariskiClosed}
###### Proposition

With [[excluded middle]] then:

Let $k$ be a [[field]], let $n \in \mathbb{N}$, and let $V(\mathcal{F}) \subset k^n$ be a Zariski closed subset (def. \ref{ZariskiOpenSubsetsOnAffineSpace}). Then the following are equivalent:

1. $V(\mathcal{F})$ is an [[irreducible closed subset]];

1. The [[vanishing ideal]] $I(V(\mathcal{F}))$ (def. \ref{VanishingIdeal}) is a [[prime ideal]].

=--

+-- {: .proof}
###### Proof

In one direction, assume that $V(\mathcal{F})$ is irreducible and consider $f,g \in k[X_1, \cdots, X_n]$ with $f \cdot g \in I(V(\mathcal{F}))$. We need to show that then already $f \in I(V(\mathcal{F}))$ or $g \in I(V(\mathcal{F}))$.

Now since $k$ is a field, we have

$$
  \left(
    f(a_1, \cdots a_n) \cdot g(a_1, \cdots, a_n) = 0
  \right)
  \Rightarrow
  \left(
    \left(
      f(a_1, \cdots, a_n) = 0
      \,\text{or}\,
      g(a_1, \cdots, a_n) = 0
    \right)
  \right)
  \,.
$$

This implies that

$$
  V(\mathcal{F}) \subset V(\{f\}) \cup V(\{g\})
$$

and hence that

$$
  V(\mathcal{F})
   =
  (V(\mathcal{F}) \cap F(\{f\}))
  \,\,\cup\,\,
  (V(\mathcal{F}) \cap F(\{g\}) )
  \,.
$$

But since $V(\{f\})$, $V(\{g\})$ and  $V(\mathcal{F})$ are all closed, by construction, their intersections are closed and hence this is a decomposition of $V(\mathcal{F})$ as a union of closed subsets. Therefore now the assumption that $V(\mathcal{F})$ is [[irreducible closed subset|irreducible]] implies that

$$
  \begin{aligned}
    &
    \left(
      \,
      V(\mathcal{F}) = V(\mathcal{F}) \cap V(\{f\})
      \,
    \right)
    \,\text{or}\,
    \left(
      \,
      V(\mathcal{F}) = V(\mathcal{F}) \cap V(\{g\})
      \,
    \right)
    \\
    \Leftrightarrow \;
    &
    \left(
       \left(
         \,
         V(\mathcal{F}) \subset V(\{f\})
         \,
       \right)
       \,\text{or}\,
       \left(
         \,
         V(\mathcal{F}) \subset V(\{g\})
         \,
       \right)
    \right)
    \\
    \Leftrightarrow \,
    &
    \left(
      \left(
        \,
        f \in I(X)
        \,
      \right)
      \,\text{or}\,
      \left(
        \,
        g \in I(X)
        \,
      \right)
    \right)
  \end{aligned}
  \,.
$$

Now for the converse, assume that $I(V(\mathcal{F}))$ is a prime ideal, and that $V(\mathcal{F}) = V(\mathcal{F}_1) \cup V(\mathcal{F}_2)$. We need to show that $V(\mathcal{F})  = V(\mathcal{F}_1)$ or that $V(\mathcal{F}) = V(\mathcal{F}_2)$.

Assume on the contrary, that there existed elements

$$
  (a_1, \cdots, a_n) \in V(\mathcal{F}_1) \backslash V(\mathcal{F}_2)
  \;\text{and}\;
  (b_1, \cdots, b_n) \in V(\mathcal{F}_2) \backslash V(\mathcal{F}_1)
  \,
$$

Then in particular the vanishing ideals would not contain each other

$$
  \not\left(
     I(V(\mathcal{F}_1))
     \subset
     I(V(\mathcal{F}_2))
  \right)
  \,\,\,\text{and}\,\,\,
  \not\left(
     I(V(\mathcal{F}_2))
     \subset
     I(V(\mathcal{F}_1))
  \right)
$$

and hence there were polynomials

$$
  f\in I(V(\mathcal{F}_1)) \backslash I(V(\mathcal{F}_2))
  \,\,\,\text{and}\,\,\,
  g \in I(V(\mathcal{F}_2)) \backslash I(V(\mathcal{F}_1))
  \,.
$$

But since a product of polynomials vanishes at some point once one of the factors vanishes at that point, it would follows that

$$
  f \cdot g \in I(V(\mathcal{F}_1)) \cap I(V(\mathcal{F}_2)) = I(V(\mathcal{F}))
  \,,
$$

which were in contradiction to the assumption that $I(V(\mathcal{F}))$ is a prime ideal. Hence we have a [[proof by contradiction]].

=--

Proposition \ref{PrimeVanishingIdealOfIrreducibleZariskiClosed} gives an [[injection]]

$$
  \left\{
    \array{
      \text{irreducible Zariski closed}
      \\
      V(\mathcal{F}) \subset k^n
    }
  \right\}
  \overset{\phantom{AAA}}{\hookrightarrow}
  \left\{
    \array{
      \text{prime ideals}
      \\
      I \in k[X_1, \cdots, X_n]
    }
  \right\}
  \,.
$$

The following says that for [[algebraically closed fields]] then this is in fact a [[bijection]]:

+-- {: .num_prop #PrimeIdealClosedsubspaceBijection}
###### Proposition

Let $k = \overline{k}$ be an [[algebraically closed field]] and  let $n \in \mathbb{N}$. Then the function

$$
  \array{
    IrrClSub(\mathbb{A}^n_k)
      &\overset{}{\longrightarrow}&
    PrimeIdl(k[X_1, \cdots, X_n])
    \\
    V(\mathcal{F})
      &\overset{\phantom{AAA}}{\mapsto}&
    I(V(\mathcal{F}))
  }
$$

from prop. \ref{PrimeVanishingIdealOfIrreducibleZariskiClosed} is a [[bijection]].

=--

The **proof** uses [[Hilbert's Nullstellensatz]].

+-- {: .num_remark}
###### Remark
**(generalization to affine varieties)**

Prop \ref{ZariskiClosedSubsetsInSpecR} suggests to consider the set of [[prime ideals]] of a [[polynomial ring]] $k[X_1, \cdots, X_n]$ for general $k$ as more fundamental, in some sense, than the set $k^n$. Morover, the set of prime ideals makes sense for every [[commutative ring]] $R$, not just $R = k[X_1, \cdots, X_n]$, and hence this suggests to consider a Zariski topology on  sets of prime ideals. This leads to the more general concept of Zariski topologies for [[affine varieties]], def. \ref{ZariskiClosedSubsetsInSpecR} below.

=--

### Examples

+-- {: .num_example #AffineSpaceOverInfiniteFieldNotHausdorff}
###### Example

If the [[field]] $k$ is _not_ a [[finite field]], then the
Zariski topology on the [[affine space]] (def. \ref{ZariskiOpenSubsetsOnAffineSpace}) is _not_ [[Hausdorff topological space|Hausdorff]].

This is because the solution set to a system of [[polynomials]] over an infinite polynomial is always a [[finite set]]. This means that in this case all the Zariski closed subsets $V(\mathcal{F})$ are [[finite sets]]. This in turn implies that the [[intersection]] of _every_ pair of [[inhabited set|non-empty]] Zariski open subsets is [[inhabited|non-empty]].

But the Zariski topology is always [[sober topological space|sober]], see prop. \ref{ZariskiTopologyIsSober} below.

=--

## On affine varieties
 {#OnAffineVarieties}

### Definition

+-- {: .num_defn #ZariskiClosedSubsetsInSpecR}
###### Definition
**(Zariski topology on set of prime ideals)**

Let $R$ be a [[commutative ring]]. Write $PrimeIdl(R)$ for its set of [[prime ideals]]. For $\mathcal{F} \subset R$ any subset of elements of the ring, consider the subsets of those prime ideals that contain $\mathcal{F}$:

$$
  V(\mathcal{F})
   \;\coloneqq\;
  \left\{
    p \in PrimeIdl(R) \,\vert\, \mathcal{F} \subset p
  \right\}
  \,.
$$

These are called the _Zariski [[closed subsets]]_ of $PrimeIdl(R)$. Their [[complements]] are called the _Zariski open subsets_.

=--

+-- {: .num_prop #WellDefinedZariskiTopologyOnSpecR}
###### Proposition
**(Zariski topology well defined)**

Assuming [[excluded middle]], then:

Let $R$ be a [[commutative ring]]. Then the collection of Zariski open subsets (def. \ref{ZariskiClosedSubsetsInSpecR}) in its set of [[prime ideals]]

$$
  \tau_{Spec(R)} \subset P(PrimeIdl(R))
$$

satisfies the axioms of a [[topological space|topology]], the _Zariski topology_.

This [[topological space]]

$$
  Spec(R) \coloneqq (PrimeIdl(R), \tau_{Spec(R)})
$$

is called (the space underlying) the _[[prime spectrum of a commutative ring|prime spectrum of the commutative ring]]_.

=--

+-- {: .proof #WellDefinedZariskiTopologyOnSpecRProof}
###### Proof

For $\mathcal{F} \subset R$ write $\mathcal{F} \subset I(\mathcal{F})$ for the [[ideal]] which is generated by $\mathcal{F}$. Evidently the Zariski closed subsets depend only on this ideal

$$
  V(I(\mathcal{F})) = V(\mathcal{F})
$$

and therefore it is sufficient to consider the $V(\mathcal{F})$ for the case that $\mathcal{F} \subset R$  is not just a subset, but an ideal.

So let $\{F_i \in Idl(R)\}_{i \in I}$ be a set of ideals in $R$ and let $\{V(\mathcal{F}_i) \subset PrimeIdl(R)\}_{i \in I}$ be the corresponding set of Zariski closed subsets. We need to show that there exists $\mathcal{F}_\cup, \mathcal{F}_\cap \subset R$ such that

1. $\underset{i \in I}{\cap} V(\mathcal{F}_i) = V(\mathcal{F}_\cup)$;

1. if $I$ is [[finite set]] then $\underset{i \in I}{\cup} V(\mathcal{F}_i) = V(\mathcal{F}_\cap)$.

We claim that

* $\mathcal{F}_{\cup} = \underset{i \in I}{\sum} \mathcal{F}_i \coloneqq \left\{ \underset{i \in I}{\sum} f_i  \,, \in R\;\vert\; f_i \in \mathcal{F}_i \right\}$

* $\mathcal{F}_{\cap} = \underset{i \in I}{\prod} \mathcal{F}_i \coloneqq \left\{ \underset{i \in I}{\prod} f_i \, \in R \;\vert\; f_i \in \mathcal{F}_i \right\}$,

Regarding the first point:

By using the various definitions, we get the following chain of logical equivalences:

$$
  \begin{aligned}
    & p \in \underset{i \in I}{\cap} V(\mathcal{F}_i)
    \\
    \Leftrightarrow\;
    &
    \underset{i \in I}{\forall}
    \left(
      p \in V(\mathcal{F}_i)
    \right)
    \\
    \Leftrightarrow\;
    &
    \underset{i \in I}{\forall}
    \left(
      \mathcal{F}_i \subset p
    \right)
    \\
    \Leftrightarrow\;
    &
    \left(\underset{i \in I}{\sum} \mathcal{F}_i\right)
    \subset p
    \\
    \Leftrightarrow\;
    &
    p \in V\left(
      \underset{i \in I}{\sum} \mathcal{F}_i
    \right)
    \,.
  \end{aligned}
$$

Regarding the second point, in one direction we have the immediate implication

$$
  \begin{aligned}
    &
    p \in \underset{i \in I}{\cup} V(\mathcal{F}_i)
    \\
    \Leftrightarrow\;
    &
    \underset{i \in I}{\exists}
    \left(
      \mathcal{F}_i \subset p
    \right)
    \\
    \Rightarrow \;
    &
    \underset{(f_i) \in \underset{i \in I}{\prod} \mathcal{F}_i}{\forall}
    \left(
      \underset{i \in I}{\prod} f_i \in p
    \right)
    \\
    \Leftrightarrow\;
    &
    p
    \in
    V\left(
      \underset{i \in I}{\prod} \mathcal{F}_i
    \right)
    \,.
  \end{aligned}
$$

For the converse direction we need to show that

$$
  \left(
    p
    \in
      V\left(
        \underset{i \in I}{\prod} \mathcal{F}_i
      \right)
   \right)
     \;\Rightarrow\;
   \left(
     p
     \in
     \underset{i \in I}{\cup} V(\mathcal{F}_1)
   \right)
  \,.
$$

hence that

$$
  \left(
    \underset{(f_i) \in \underset{i \in I}{\prod} \mathcal{F}_i }{\forall}
    \left(
      \underset{i \in I}{\prod} f_i \in p
    \right)
  \right)
  \;\Rightarrow\;
  \left(
    \underset{i \in I}{\exists}
    \left(
      \underset{f_i \in \mathcal{F}_i}{\forall}
      \left(
         f_i \in p
      \right)
    \right)
  \right)
  \,.
$$

By [[excluded middle]], this is equivalent to its [[contraposition]], which by [[de Morgan's law]] is

$$
  \left(
    \underset{i \in I}{\forall}
    \left(
      \underset{f_i \in \mathcal{F}_i}{\exists}
      \not
      \left(
         f_i \in p
      \right)
    \right)
  \right)
  \;\Rightarrow\;
  \left(
    \underset{(f_i) \in \underset{i \in I}{\prod} \mathcal{F}_i }{\exists}
    \not
    \left(
      \underset{i \in I}{\prod} f_i \in p
    \right)
  \right)
  \,.
$$

This holds by the assumption that $p$ is a [[prime ideal]].

=--

### Properties
  {#PropertiesOnAffineVarieties}

We discuss some properties of the Zariski topology on [[prime spectra of commutative rings]].

#### Topological closures

+-- {: .num_lemma #ZariskiClorsuredOfPont}
###### Lemma
**([[topological closure]] of points)**

Let $R$ be a [[commutative ring]] and consider $Spec(R) = (PrimeIdl(R), \tau_{Spec(R)})$ its [[prime spectrum of a commutative ring|prime spectrum]] equipped with the Zariski topology (def. \ref{ZariskiClosedSubsetsInSpecR}).

Then the [[topological closure]] of a point $p \in PrimeIdl(R)$ is $V(p) \subset PrimeIdl(R)$ (def. \ref{ZariskiClosedSubsetsInSpecR}).

=--

+-- {: .proof}
###### Proof

By definition the topological closure of $\{p\}$ is

$$
  Cl(\{p\})=\underset{ {I \in Idl(R) } \atop { p \in V(I) }  }{\cap} V(I)
  \,.
$$

Hence unwinding the definitions, we have the following sequence of logical equivalences:

$$
  \begin{aligned}
    & q \in Cl(\{q\})
    \\
    \Leftrightarrow\; &
    q \in
    \underset{ {I \in Idl(R)} \atop { p \in V(I) }   }{\cap} V(I)
    \\
    \Leftrightarrow\;
    &
    \underset{ { I \in Idl(R) } \atop { I \subset p } }{\forall} (q \in V(I))
    \\
    \Leftrightarrow\;
    &
    \underset{ { I \in Idl(R) } \atop { I \subset p } }{\forall} (I \subset q)
    \\
    \Leftrightarrow\;
    &
    p \subset q
    \\
    \Leftrightarrow\;
     &
    q \in V(p)
  \end{aligned}
$$

=--

Recall:

+-- {: .num_lemma #PrimeIdealTheorem}
###### Lemma
**([[prime ideal theorem]])

Assuming the [[axiom of choice]] or at least the [[ultrafilter principle]] then:

For $R$ a [[commutative ring]] and $I \subset R$ a [[proper ideal]], then $I$ is contained in some [[prime ideal]].

=--

The [[axiom of choice]] even implies that every proper ideal is contained in a [[maximal ideal]] (by [this prop.](maximal+ideal#EveryProperIdealisContainedInAMaximalOne)).

+-- {: .num_prop #MaximalIdealsAreClosedPoints}
###### Proposition
**([[maximal ideals]] are [[closed points]])

Let $R$ be a [[commutative ring]], consider the [[topological space]] $Spec(R) = (PrimeIdl(R),\tau_{Spec(R)})$, i.e. its [[prime spectrum of a commutative ring|prime spectrum]] equipped with the Zariski topology from def. \ref{ZariskiClosedSubsetsInSpecR}.

Then the [[maximal ideals]] inside the prime ideals constitute [[closed points]].

Assuming the [[axiom of choice]] or at least the [[ultrafilter principle]] then also the converse is true:

Then the inclusion of [[maximal ideals]] $\mathfrak{m} \in MaxIdl(R) \subset PrimeIdl(R)$ into all [[prime ideals]] is precisely the inclusion of the subset of [[closed points]] into all points of $Spec(R)$.

$$
  ClosedPoints(Spec(R)) \simeq MaxIdl(R) \subset PrimeIdl(R)
  \,.
$$

=--

+-- {: .proof}
###### Proof

By lemma \ref{ZariskiClorsuredOfPont} we have

$$
  Cl(\{p\}) = V(p)
$$

and hence we need to show that

$$
  \{\mathfrak{m}\} = V(\mathfrak{m})
$$

precisely if $\mathfrak{m}$ is maximal.

In one direction, assume that $\mathfrak{m}$ is maximal. By definition $V(\mathfrak{m})$ contains all the prime ideals $p$ such that $\mathfrak{m} \subset p$. That $\mathfrak{m}$ is maximal means that it is not contained in a larger proper ideal, in particular not in any larger prime ideal, and hence $V(\mathfrak{m}) = \{\mathfrak{m}\}$.

In the other direction, assume that $\mathfrak{m}$ is a prime ideal such that $V(\mathfrak{m}) = \{\mathfrak{m}\}$. By definition this means equivalently that the only prime ideal $p$ with $\mathfrak{m} \subset p$ is $\mathfrak{m}$ itself. We need to show that more generally $\mathfrak{m} \subset I$ for $I$ any [[proper ideal]] implies that $\mathfrak{m} = I$.

But the [[axiom of choice]]/[[ultrafilter principle]] imply the [[prime ideal theorem]] (lemma \ref{PrimeIdealTheorem}), which says that there is a prime ideal $p$ with $I \subset p$, hence a sequence of inclusions $\mathfrak{m} \subset I \subset p$. Since this implies $\mathfrak{m} \subset p$, we have $\mathfrak{m} = p$, hence $I = \mathfrak{m}$.

=--

#### Irreducible closed subsets as prime ideals

+-- {: .num_prop #ZariskiIrreducibleClosedSubsetsArePreciselyPrimeIdeals}
###### Proposition
**([[irreducible closed subsets]] correspond to [[prime ideals]])**

With [[excluded middle]] then:

Let $R$ be a [[commutative ring]], and let $\mathcal{F} \subset R$ be an ideal in $R$, hence $V(\mathcal{F}) \subset Spec(R)$ is a Zariski closed subset in the [[prime spectrum of a commutative ring|prime spectrum]] of $R$. Then the following are equivalent:

1. $V(\mathcal{F})$ is an [[irreducible closed subset]];

1. $\mathcal{F}$ is a [[prime ideal]].

=--

+-- {: .proof}
###### Proof

In one direction, assume that $V(\mathcal{F})$ is irreducible, and that $f,g \in R$ with $f \cdot g \in \mathcal{F}$. We need to show that then already $f \in \mathcal{F}$ or $g \in \mathcal{F}$.

To this end, first observe that

$$
  V(\mathcal{F}) \subset V((f)) \cup V((g))
  \,.
$$

This is because

$$
  \begin{aligned}
    & p \in V(\mathcal{F})
    \\
    \Leftrightarrow\;
    &
    \mathcal{F} \subset p
    \\
    \Rightarrow \;
    &
    f \cdot g \in p
    \\
    \Rightarrow\;
    &
    \left(
      f \in p
    \right)
    \,\text{or}\,
    \left(
      g \in p
    \right)
    \\
    \Leftrightarrow\;
    &
    \left(
      p \in V(g)
    \right)
    \,\text{or}\,
    \left(
      p \in V(f)
    \right)
    \\
    \Leftrightarrow\;
    &
    p \in V(f) \cup V(g)
    \,,
  \end{aligned}
$$

where the implication in the middle uses that $p$ is a prime ideal.

It follows that

$$
  V(\mathcal{F})
  \;=\;
  \left(
    V(f) \cap V(\mathcal{F})
  \right)
   \cup
  \left(
    V(g) \cap V(\mathcal{F})
  \right)
  \,.
$$

This is a decomposition of $V(\mathcal{F})$ as a union of closed subsets,
hence the assumption that $V(\mathcal{F})$ is irreducible implies that

$$
  \begin{aligned}
     &
     \left(
       V(\mathcal{F}) = V(f) \cap V(\mathcal{F})
     \right)
     \,\text{or}\,
     \left(
        V(\mathcal{F}) = V(g) \cap V(\mathcal{F})
     \right)
     \\
     \Leftrightarrow
     &
     \left(
        V(\mathcal{F})
        \subset
        V(f)
     \right)
     \,\text{or}\,
     \left(
       V(\mathcal{F}) \subset V(g)
     \right)
     \\
     \Leftrightarrow\,
     &
     \left(
         f \in \mathcal{F}
     \right)
     \,\text{or}\,
     \left(
        g \in \mathcal{F}
     \right)
     \,.
  \end{aligned}
$$

Now for the converse. Assume that $\mathcal{F}$ is a prime ideal and that $V(\mathcal{F}) = V(\mathcal{F}_1) \cup V(\mathcal{F}_2)$. Observe (as in the [proof](#WellDefinedZariskiTopologyOnSpecRProof) of prop. \ref{WellDefinedZariskiTopologyOnSpecR}) that this means equivalently that $\mathcal{F} = \mathcal{F}_1 \cdot \mathcal{F}_2$.
We need to show
that then $V(\mathcal{F}) = V(\mathcal{F}_1)$ or that $V(\mathcal{F} = V(\mathcal{F}_2))$.

Suppose on the contrary that neither $\mathcal{F}_1$ nor $\mathcal{F}_2$ coincided with $\mathcal{F}$. This means that there were elements $f \in \mathcal{F}_1 \backslash \mathcal{F}$ and $g \in \mathcal{F}_2 \backslash \mathcal{F}$ such that still $f \cdot g \in \mathcal{F}$, in contradiction to the assumption. Hence we have a [[proof by contradiction]].

=--

As a corollary:

+-- {: .num_prop #ZariskiTopologyIsSober}
###### Proposition
**([[schemes are sober|Zariski topology on prime spectra is sober]])**

With [[excluded middle]] and [[axiom of choice]] (or at least the [[ultrafilter principle]]) then:

Let $R$ be a [[commutative ring]]. Then $Spec(R)$ (its [[prime spectrum of a commutative ring|prime spectr]] equipped with the Zariski topology of def. \ref{ZariskiClosedSubsetsInSpecR}) is a [[sober topological space]].

=--

+-- {: .proof}
###### Proof

We need to show that the function

$$
  Cl(\{-\})
   \;\colon\;
  PrimeIdl(R)
    \longrightarrow
  IrrClSub(Spec(R))
$$

which sends a point to its topological closure, is a [[bijection]].

By lemma \ref{ZariskiClorsuredOfPont} this function is given by sending a [[prime ideal]] $p \in PrimeIdl(R)$ to the Zariski closed subset $V(p)$. That this is a bijection is the statement of prop. \ref{ZariskiIrreducibleClosedSubsetsArePreciselyPrimeIdeals}.

=--

### Examples

+-- {: .num_example #AffinSpaceAsPrimeSpectrum}
###### Example
**(affine space as prime spectrum)**

Reconsider the case where $R = k[X_1,\cdots, X_n]$
is a [[polynomial ring]], for $k$ a [[field]], as in the discussion of the naive affine space $k^n$
[above](#OnAffineSpace).

Observe that, by \ref{MaximalIdealsAreClosedPoints}, the [[closed points]] in the [[prime spectrum of a commutative ring|prime spectrum]] $Spec(k[X_1, \cdots, X_n])$ correspond to the [[maximal ideals]] in the [[polynomial ring]]. These are of the form

$$
  (a_1, \cdots, a_n)
  \coloneqq
  \left(
   (X_1 - a_1) \cdot (X_2 - a_2) \cdots (X_n - a_n)
  \right)
$$

and hence are in bijection with the points of the naive affine space

$$
  k^n \simeq MaxIdl(k[X_1, \cdots, X_n])
  \,.
$$

There is however also [[prime ideals]] in $k[X_1, \cdots, X_n]$ which are not maximal. In particular there is the 0-ideal $(0)$.

=--

+-- {: .num_prop #SpecZ}
###### Proposition
**([[Spec(Z)]])**

Let $R = \mathbb{Z}$ be the [[commutative ring]] of [[integers]]. Consider the corresponding Zariski [[prime spectrum of a commutative ring|prime spectrum]] (prop. \ref{WellDefinedZariskiTopologyOnSpecR}) [[Spec(Z)]].

The [[prime ideals]] of the ring of integers are

1. the ideals $(p)$ generated by [[prime numbers]] $p$ (this special case is what motivates the terminology "prime ideal");

1. the ideal $(0) = \{0\}$.

$$
  PrimeIdl(\mathbb{Z})
  =
  \left\{
    0, \; 2, 3, 5, 7, 11, \cdots
  \right\}
  \,.
$$

All the prime ideals $p \geq 2$ are [[maximal ideals]]. Hence by prop. \ref{MaximalIdealsAreClosedPoints} these are [[closed points]] of $Spec(\mathbb{Z})$.

Only the prime ideal $(0)$ is not maximal, hence the point $(0)$ is not closed.
Its closure is the entire space

$$
  Cl(\{0\}) = Spec(\mathbb{Z})
  \,.
$$

To see this, notice that in fact $Spec(\mathbb{Z})$ is the only closed subset containing the point $(0)$. This is because

$$
  \begin{aligned}
    & (0) \in V(I)
    \\
    \Leftrightarrow\;
    &
    I \subset (0)
    \\
    \Leftrightarrow\;
    &
    I = (0)
  \end{aligned}
$$

and $V(0) = Spec(\mathbb{Z})$, because

$$
  (p \in V(0))
  \Leftrightarrow
  (0 \subset p)
  \Leftrightarrow
  true
  \,.
$$

=--

## In terms of Galois connections
 {#InTermsOfGaloisConnections}

We now discuss how all of the above constructions and statements, and a bit more, follows immediately as a special case of the general theory of what is called _[[Galois connections]]_ or _[[adjoint functors]] between [[posets]]_.

### Background on Galois connections
 {#BackgroundOnGaloisConnections}

+-- {: .num_defn #GaloisConnection}
###### Definition
**([[Galois connection]] induced from a [[relation]])**

Consider two [[sets]] $X,Y \in Set$ and a [[relation]]

$$
  E \hookrightarrow X \times Y
  \,.
$$

Define two [[functions]] between their [[power sets]] $P(X), P(Y)$, as follows.
(In the following we write $E(x, y)$ to abbreviate the formula $(x, y) \in E$.)

1. Define

   $$
     V_E
       \;\colon\;
     P(X)
       \longrightarrow
     P(Y)
   $$

   by

   $$
     V_E(S)
       \coloneqq
     \left\{
       y \in Y \vert  \underset{x \in  X}{\forall} \left( \left(x \in S\right) \Rightarrow E(x, y)
    \right) \right\}
   $$

1. Define

   $$
     I_E
       \;\colon\;
     P(Y)
       \longrightarrow
     P(X)
   $$

   by

   $$
     I_E(T) \coloneqq \left\{x \in X \vert \underset{y \in Y}{\forall} \left( \left(y \in T \right) \Rightarrow E(x, y)  \right)\right\}
   $$

=--

+-- {: .num_prop #GaloisConnectionAsAdjunction}
###### Proposition

The construction in def. \ref{GaloisConnection} has the following properties:

1. $V_E$ and $I_E$ are [[contravariant functor|contravariant]] order-preserving in that

   1. if $S \subset S'$, then $V_E(S') \subset V_E(S)$;

   1. if $T \subset T'$, then $I_E(T') \subset I_E(T)$

1. The _[[adjunction]] law_ holds:
   $
     \left(
        T \subset V_E(S)
     \right)
       \,\LeftRightarrow\,
     \left(
       S \subset I_E(T)
     \right)
   $

   which we denote by writing

   $$
     P(X)
       \underoverset{\underset{V_E}{\longrightarrow}}{\overset{I_E}{\longleftarrow}}{\bot}
     P(Y)^{op}
   $$

1. both $V_E$ as well as $I_E$ take [[unions]] to [[intersections]].

=--

+-- {: .proof}
###### Proof

Regarding the first point: the larger $S$ is, the more conditions that are placed on $y$ in order to belong to $V_E(S)$, and so the smaller $V_E(S)$ will be.

Regarding the second point:  This is because both these conditions are equivalent to the condition $S \times T \subset E$.

Regarding the third point: Observe that in a poset such as $P(Y)$, we have that $A = B$ iff for all $C$, $C \leq A$ iff $C \leq B$ (this is the [[Yoneda lemma]] applied to posets). It follows that

$$
  \array{
    T \subset V_E(\bigcup_{i \in I} S_i) & iff & \bigcup_{i: I} S_i \subset I_E(T) \\
    & iff & \forall_{i: I} S_i \subset I_E(T) \\
    & iff & \forall_{i: I} T \subset V_E(S_i) \\
    & iff & T \subset \bigcap_{i: I} V_E(S_i)
}
$$

and we conclude $V_E(\bigcup_{i: I} S_i) = \bigcap_{i: I} V_E(S_i)$ by the [[Yoneda lemma]].

=--

+-- {: .num_prop #GaloisClosureOperator}
###### Proposition
**([[closure operators]] from [[Galois connection]])**

Given a [[Galois connection]] as in def. \ref{GaloisConnection}, consider the [[composition|composites]]

$$
  I_E \circ V_E \;\colon\; P(X) \longrightarrow P(X)
$$

and

$$
  V_E \circ I_E \;\colon\; P(Y) \longrightarrow P(Y)
  \,.
$$

These satisfy:

1. For all $S \in P(X)$ then  $S \subset I_E \circ V_E(S)$.

1. For all $S \in P(X)$ then $V_E \circ I_E \circ V_E (S)  =  V_E(S)$.

1. $I_E \circ V_E$ is [[idempotent]] and [[covariant functor|covariant]].

and

1. For all $T \in P(Y)$ then $T \subset V_E \circ I_E(T)$.

1. For all $T \in P(Y)$ then $I_E \circ V_E \circ I_E (T) = I_E(T)$.

1. $V_E \circ I_E$ is [[idempotent]] and [[covariant functor|covariant]].

This is summarized by saying that
$I_E \circ V_E$ and $V_E \circ I_E$ are _[[closure operators]]_ ([[idempotent monads]]).

=--

+-- {: .proof}
###### Proof

The first statement is immediate from the adjunction law (prop. \ref{GaloisConnectionAsAdjunction}).

Regarding the second statement:
This holds  because applied to sets $S$ of the form $I_E(T)$, we see $I_E(T) \subset I_E \circ V_E \circ I_E(T)$. But applying the contravariant map $I_E$ to the inclusion $T \subset V_E \circ I_E(T)$, we also have $I_E \circ V_E \circ I_E(T) \subset I_E(T)$.

This directly implies that the function $I_E \circ V_E$. is idempotent, hence the third statement.

The argument for $V_E \circ I_E$ is directly analogous.

=--

In view of prop. \ref{GaloisClosureOperator} we say that:

+-- {: .num_defn #GaloisClosedElements}
###### Definition
**(closed elements)**

Given a [[Galois connection]] as in def. \ref{GaloisConnection}, then

1. $S \in P(X)$ is called *closed* if $I_E \circ V_E(S) = S$;

1. the _closure_ of $S \in P(X)$ is $Cl(S) \coloneqq I_E \circ V_E(S)$

and similarly

1. $T \in P(Y)$ is called *closed* if $V_E \circ I_E(T) = T$;

1. the _closure_ of $T \in P(Y)$ is $Cl(T) \coloneqq V_E \circ I_E(T)$.

=--

It follows from the properties of [[closure operators]], hence form prop. \ref{GaloisClosureOperator}:

+-- {: .num_prop #GaloisFixedPoints}
###### Proposition
**([[fixed point of an adjunction|fixed points]] of a [[Galois connection]])**

Given a [[Galois connection]] as in def. \ref{GaloisConnection},
then

1. the closed elements of $P(X)$ are precisely those in the [[image]] $im(I_E)$ of $I_E$;

1. the closed elements of $P(Y)$ are precisely those in the [[image]] $im(V_E)$ of $V_E$.

We says these are the _[[fixed point of an adjunction|fixed points]]_ of the Galois connection. Therefore the restriction of the Galois connection

$$
  P(X)
    \underoverset{\underset{V_E}{\longrightarrow}}{\overset{I_E}{\longleftarrow}}{\bot}
  P(Y)^{op}
$$

to these fixed points yields an [[equivalence of categories|equivalence]]

$$
  im(I_E)
    \underoverset{\underset{V_E}{\longrightarrow}}{\overset{I_E}{\longleftarrow}}{\simeq}
  im(V_E)^{op}
$$

now called a _[[Galois correspondence]]_.

=--

+-- {: .num_prop}
###### Proposition

Given a [[Galois connection]] as in def. \ref{GaloisConnection}, then the sets of closed elements according to def. \ref{GaloisClosedElements} are closed under forming [[intersections]].

=--

+-- {: .proof}
###### Proof

If $\{T_i \in P(Y)\}_{i: I}$ is a collection of elements closed under the operator $K = V_E \circ I_E$, then by the first item in prop. \ref{GaloisClosureOperator} it is automatic that $\bigcap_{i: I} T_i \subset K(\bigcap_{i: I} T_i)$, so it suffices to prove the reverse inclusion. But since $\bigcap_{i: I} T_i \subset T_i$ for all $i$ and $K$ is covariant and $T_i$ is closed, we have $K(\bigcap_{i: I} T_i) \subset K(T_i) \subset T_i$ for all $i$, and $K(\bigcap_{i: I} T_i) \subset \bigcap_{i: I} T_i$ follows.

=--

### Applied to affine space
  {#GaloisConnectionAppliedToAffineSpace}

We now redo the discussion of the Zariski topology on the affine space $k^n$ from [above](#OnAffineSpace) as a special case of the general considerations of [[Galois connections]].

+-- {: .num_example #ZariskiClosedSubsetsInaffineViaGalois}
###### Example
**(Zariski closed subsets in affine space via Galois connection)**

Let $k$ be a [[field]] and let $n \in \mathbb{N}$, and write $k[X_1, \cdots, X_n]$ for the [[polynomial ring]] over $k$  in $n$ [[variables]].  Define a [[relation]]

$$
  E \hookrightarrow k[x_1, \ldots, x_n] \times k^n
$$

by

$$
  E(f, x)\coloneqq \left( f(x) = 0\right)
  \,.
$$

By def. \ref{GaloisConnection} and prop. \ref{GaloisConnectionAsAdjunction} we obtain the corresponding [[Galois connection]] of the form

$$
  P(k[X_1, \cdots, X_n])
     \underoverset{\underset{V_E}{\longrightarrow}}{\overset{I_E}{\longleftarrow}}{\bot}
  P(k^n)^{op}
$$

(where now $k[X_1, \cdots, X_n]$ and $k^n$ denote their underlying sets).

Here by def. \ref{GaloisConnection} the function

$$
  V_E \;\colon\; P(k[x_1, \ldots, x_n]) \longrightarrow P(k^n)
$$

sends a set $\mathcal{F}$ of [[polynomials]]  to its corresponding *[[variety]]*,

$$
  V_E(\mathcal{F}) = \{\vec x \in k^n \,\vert\, \forall_{f \in k[x_1, \ldots, x_n]} \; (f \in \mathcal{F}) \Rightarrow (f(x) = 0)\}
  \,.
$$

These are just the Zariski closed subsets from def. \ref{ZariskiOpenSubsetsOnAffineSpace}.

In the other direction,

$$
  I_E \;\colon\; P(k^n) \longrightarrow P(k[x_1, \ldots, x_n])
$$

sends a set of points $T \subseteq k^n$ to its corresponding *[[vanishing ideal]]*

$$
  I_E(T) = \{f \in k[x_1, \ldots, x_n] \,\vert\, \forall_{x: k^n} \; x \in T \Rightarrow f(x) = 0\}
$$

which we considered earlier in def. \ref{VanishingIdeal}.

=--

We may now use the abstract theory of Galois connections to verify that Zariski closed subsets form a [[topological space|topology]]:

+-- {: .num_prop #ZariskiTopologyOnAffineSpaceViaGaloisConnectionWellDefined}
###### Proposition
**(Zariski topology is well defined)**

Using [[excluded middle]], then:

The set of Zariski closed subsets of $k^n$ from example \ref{ZariskiClosedSubsetsInaffineViaGalois} constitutes a [[topological space|topology]] in that it is closed under

1. arbitrary intersections;

1. finite untions.

=--

+-- {: .proof}
###### Proof

Regarding the first point: From prop. \ref{GaloisConnectionAsAdjunction} we know that $V_E$ takes unions to intersections, hence that

$$
  \underset{i \in I}{\cap} V_E(\mathcal{F}_i)
  \;=\;
  V_E\left(
    \underset{i \in I}{\cup} \mathcal{F}_i
  \right)
  \,.
$$

Regarding the second point, we exploit the [[commutative ring]] structure of $k[x_1, \ldots, x_n]$. It is sufficient to show that the set of Zariski closed sets is closed under the empty union and under binary unions.

The empty union is the entire space $k^n$, which is $V(1)$ (the variety associated with the constant polynomial $1$),

Hence it only remains to see closure under binary unions.

To this end, recall from prop. \ref{GaloisClosureOperator} that we may replace $\mathcal{F}$ with the corresponding ideal

$$
  I \coloneqq I_E \circ V_E(\mathcal{F})
$$

without changing the variety:

$$
  V_E(I) = V_E(\mathcal{F})
  \,.
$$

With this it is sufficient to show that

$$
  V_E(I) \cup V_E(I') = V(I \cdot I')
$$

where $I \cdot I'$ is the ideal consisting of finite sums of elements of the form $f g$ with $f \in I$ and $g \in I'$.

We conclude by proving this statement:

Applying the contravariant operator $V_E$ to the inclusions $I \cdot I' \subseteq I$ and $I \cdot I' subseteq I'$ (which are clear since $I, I'$ are ideals), we derive $V_E(I) \subseteq V_E(I \cdot I')$ and $V_E(I') \subseteq V(I \cdot I')$, so the inclusion $V_E(I) \cup V_E(I') \subseteq V(I \cdot I')$ is automatic.

In the other direction, to prove $V(I \cdot I') \subseteq V_E(I) \cup V(I')$, suppose $x \in V(I \cdot I')$ and that $x$ *doesn't* belong to $V(I)$. Then $f(x) \neq 0$ for some $f \in I$. For every $g \in I'$, we have $f(x)g(x) = (f \cdot g)(x) = 0$ since $f \cdot g \in I \cdot I'$ and $x \in V_E(I \cdot I')$. Now divide by $f(x)$ to get $g(x) = 0$ for every $g \in I'$, so that $x \in V_E(I')$.

=--

+-- {: .num_example #ZariskiTopologyOnMaximalIdealsOfPolynomialRingViaGaloisConnection}
###### Example

Let $k$ be a [[field]], let $n \in \mathbb{N}$ and write $k[X_1, \cdots, X_n]$ for the [[polynomial ring]] over $k$ in $n$ [[variables]], and $MaxIdl(k[X_1, \cdots, X_n])$ for the [[set]] of [[maximal ideals]] in this ring.

Define then a [[relation]]

$$
  E \hookrightarrow k[x_1, \ldots, x_n] \times MaxIdeal(k[x_1, \ldots, x_n])
$$

by

$$
  E(f, M) \Leftrightarrow (f \in M)
  \,.
$$

For a subset $T \subseteq MaxIdl(k[x_1, \ldots, x_n])$ we calculate

$$
  I_E(T) = \{f \in k[x_1, \ldots, x_n]: \forall_{\mathfrak{m} \in MaxIdl} M \in S \Rightarrow f \in \mathfrak{m}\} = \bigcap_{\mathfrak{m} \in S} \mathfrak{m}
$$

which is an ideal, since the intersection of any collection of ideals is again an ideal. (However, not all ideals are given as intersections of maximal ideals, a point to which we will return in a moment.)

=--

+-- {: .num_remark}
###### Remark

This is a slight generalization of example \ref{ZariskiClosedSubsetsInaffineViaGalois} since each point $a = (a_1, \ldots, a_n)$ induces a maximal ideal

$$
  \mathfrak{m}_a \coloneqq \langle x_1 - a_1, \ldots, x_n - a_n \rangle
  \,,
$$

i.e. the [[kernel]] of the function

$$
  \array{
    k[x_1, \ldots, x_n] &\longrightarrow& k
    \\
    f &\mapsto& f(a)
  }
$$

which evaluates polynomials $f$ at the point $a$, where we have $f(a) = 0$ iff $f \in \mathfrak{m}_a$.

Of course it need not be the case that all maximal ideals $\mathfrak{m}$ are given by points in this way; for example, the ideal $(x^2 + 1)$ is maximal in $\mathbb{R}[x]$ but is not given by evaluation at a point because $x^2 + 1$ does not vanish at any real point. However, if the [[ground field]] $k$ is [[algebraically closed field|algebraically closed]], then every maximal ideal of $k[x_1, \ldots, x_n]$ is given by evaluation at a point $a = (a_1, \ldots, a_n)$. This result is not completely obvious; it is sometimes called the _weak [[Nullstellensatz]]_.

=--

+-- {: .num_prop}
###### Proposition

The set $S \subseteq k^n$ that are closed under the operator $V_E \circ I_E: P(k^n) \to P(k^n)$ in example \ref{ZariskiTopologyOnMaximalIdealsOfPolynomialRingViaGaloisConnection} form a [[topology]].

=--

+-- {: .proof}
###### Proof

The proof is virtually the same as in the proof of prop. \ref{ZariskiTopologyOnAffineSpaceViaGaloisConnectionWellDefined}: they are closed under arbitrary intersections by our earlier generalities, and they are closed under finite unions by the similar reasoning: $V_E(S) = V_E(I)$ where $I = I_E \circ V_E(S)$ is an ideal, so there is no loss of generality in considering $V_E(I)$ for ideals $I$, and $V_E(I) \cup V_E(I') = V_E(I \cdot I')$. If $\mathfrak{m} \in V_E(I \cdot I')$ (meaning $I \cdot I' \subseteq M$) but $\mathfrak{m}$ *doesn't* belong to $V_E(I)$, i.e., $f \notin \mathfrak{m}$ for some $f \in I$, then for every $g \in I'$ we have $f \cdot g \in \mathfrak{m}$. Taking the [[quotient]] map $\pi: R \to R/\mathfrak{m}$ to the field $R/\mathfrak{m}$, we have $\pi(f \cdot g) = \pi(f)\cdot \pi(g) = 0$, and since $\pi(f) \neq 0$ we have $\pi(g) = 0$ for every $g \in I'$, hence $\mathfrak{m} \in V_E(I')$.

=--

Thus the fixed elements of $V_E \circ I_E$ on one side of the Galois correspondence are the closed sets of a topology. The fixed elements of $I_E \circ V_E$ on the other side are a matter of interest; in the case where $k$ is [[algebraically closed field|algebraically closed]], they are the *[[radical ideals]]* of $k[X_1, \ldots, X_n]$ according to the "strong" [[Nullstellensatz]].

### Applied to affine schemes
 {#GaloisAppliedToAffineSchemes}

We now redo the discussion of the Zariski topology on the [[prime spectrum of a commutative ring]] from [above](#OnAffineVarieties) as a special case of the general considerations of [[Galois connections]].

## Related concepts

* [[Zariski site]]

* [[schemes are sober]]

## References

Lecture notes include

* Jim Carrell, _Zariski topology_ [pdf](https://personal.math.ubc.ca/~carrell/423.pdf)

See also

* Wikipedia, _[Zariski topology](https://en.wikipedia.org/wiki/Zariski_topology)_

[[!redirects Zariski topologies]]


Zdenek Dolezal

* [personal page](http://ipnp.cz/~dolezal/)

## Selected writings

On [[B meson]]-[[decays]] and [[flavour anomalies]]:

* [[Patrick Koppenburg]], [[Zdenek Dolezal]], [[Maria Smizanska]], _Rare decays of b hadrons_, Scholarpedia, 11(6):32643 ([doi:10.4249/scholarpedia.32643](http://dx.doi.org/10.4249/scholarpedia.32643))

category: people


zekeriya arvasi

Zekeriya Arvasi is a Turkish mathematician with expertise in category theory, especially simplicial ojects, crossed modules, algebraic models for homotopy connected types. In homotopy theory he collaborated  with E. Ulualan.

[Home Page](http://fef.ogu.edu.tr/zarvasi/)


Zena M. Ariola

* [personal page](http://ix.cs.uoregon.edu/~ariola/)

* [institute page](https://cas.uoregon.edu/directory/profiles/all/ariola)

## Selected writings

Introduction to [[proof theory]] via [[natural deduction]], [[sequent calculus]]:

* [[Paul Downen]], [[Zena M. Ariola]], *A tutorial on computational classical logic and the sequent calculus*, Journal of Functional Programming **28** (2018) e3 [[doi:10.1017/S0956796818000023](https://doi.org/10.1017/S0956796818000023), [pdf](https://www.pauldownen.com/publications/sequent-intro.pdf)]

category: people


Zeno

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--

An ancient [[philosophy|philosopher]] known from [[Plato]]'s [[Parmenides dialogue]], renowned for his [[paradoxes]] (which [[Hegel]] claims to be the origin of [[dialectics]] ) such as [[Zeno's paradox of motion]].

## As seen by Hegel

[[Georg Hegel]], in _[Lectures on the History of Philosophy -- Zeno](Lectures+on+the+History+of+Philosophy)_ writes this:

> {#HegelHistory1} What specially characterizes Zeno is the [[dialectic]] which, properly speaking, begins with him;

> he is the master of the Eleatic school in whom its pure thought arrives at the movement of the Notion in itself and becomes the pure soul of science. That is to say, in the Eleatics hitherto considered, we only have the proposition: "The nothing has no reality and is not at all, and thus what is called origin and decease disappears." With Zeno, on the contrary, we certainly see just such an assertion of the one and removal of what contradicts it, but we also see that this assertion is not made the starting point; for reason begins by calmly demonstrating in that which is established as existent, its negation. Parmenides asserts that "The all is immutable, for, in change, the non-being of that which is would be asserted, but Being only is; in saying that "non-being is, the subject and the predicate contradict themselves." Zeno, on the other hand, says: "Assert your change; in it as change there is the negation to it, or it is nothing." To the former change existed as motion, definite and complete. Zeno protested against motion as such, or pure motion.

> Pure Being is not motion; it is rather the negation of motion." We find it specially interesting that there is in Zeno the higher consciousness, the consciousness that when one determination is denied, this negation is itself again a determination, and then in the absolute negation not one determination, but both the opposites must be negated. Zeno anticipated this, and because he foresaw that Being is the opposite of nothing, he denied of the One what must be said of the nothing. But the same thing must occur with all the rest. We find this higher dialectic in Plato's Parmenides; here it only breaks forth in respect to some determinations, and not to the determination of the One and of Being. The higher consciousness is the consciousness of the nullity of Being as of what is determined as against the nothing, partly found in Heraclitus and then in the Sophists; with them it never has any truth, it has no existence in itself, but is only the for-another, or the assurance of the individual consciousness, and assurance as refutation, i.e. the negative side, of dialectic.

> According to Diogenes Laërtius, (IX. 25) Zeno was like wise an Eleat; he is the youngest, and lived most in company with Parmenides. The latter became very fond of him and adopted him as a son; his own father was called Telentagoras. Not in his State alone was his conduct held in high respect, for his fame was universal, and he was esteemed particularly as a teacher. Plato mentions that men came to him from Athens and other places, in order to profit from his learning. Proud self-sufficiency is ascribed to him by Diogenes (IX. 28) because he — with the exception of a journey made to Athens — continued to reside in Elea, and did not stay a longer time in the great, mighty Athens, and there attain to fame. In very various narratives his death was made for ever celebrated for the strength of his mind evinced in it; it was said that he freed a State (whether his own home at Elea or in Sicily, is not known) from its Tyrant (the name is given differently, but an exact historical account has not been recorded) in the following way, and by the sacrifice of his life. He entered into a plot to overthrow the Tyrant, but this was betrayed. When the Tyrant now, in face of the people, caused him to be tortured in every possible way to get from him an avowal of his confederates, and when he questioned him about the enemies of the State, Zeno first named to the Tyrant all his friends as participators in the plot, and then spoke of the Tyrant himself as the pest of the State. The powerful remonstrances or the horrible tortures and death of Zeno aroused the citizens, inspired them with courage to fall upon the Tyrant, kill him, and liberate themselves. The manner of the end, and his violent and furious state of mind, is very variously depicted. He is said to have pretended to wish to say something into the Tyrant's ear, and then to have bitten his ear, and thus held him fast until lie was slain by the others. Others say that he seized him by the nose between his teeth; others that as on his reply great tortures were applied, he bit off his tongue and spat it into the Tyrant's face, to show him that he could get nothing from him, and that he then was pounded in a mortar.

> It has just been noticed that Zeno had the very important character of being the originator of the true objective dialectic. Xenophanes, Parmenides, and Melisibus, start with the proposition: "Nothing is nothing; the nothing does not exist at all, or the like is real existence," that is, they make one of the opposed predicates to be existence. Now when they encounter the opposite in a determination, they demolish this determination, but it is only demolished through another, through my assertion, through the distinction that I form, by which one side is made to be the true, and the other the null. We have proceeded from a definite proposition; the nullity of the opposite does not appear in itself; it is not that it abrogates itself, i.e. that it contains a contradiction in itself. For instance, I assert of something that it is the null; then I show this by hypothesis in motion, and it follows that it is the null. But another consciousness does not assert this I declare one thing to be directly true; another has the right of asserting something else as directly true, that is to say, motion. Similarly what seems to be the case when one philosophic system contradicts another, is that the first is pre-established, and that men starting from this point of view, combat the other. The matter is thus easily settled by saying: "The other has no truth, because it does not agree with me," and the other bas the right to say the same. It does not help if I prove my system or my proposition and then conclude that thus the opposite is false; to this other proposition the first always seems to be foreign and external. Falsity must not be demonstrated through another, and as untrue because the opposite is true, but in itself; we find this rational perception in Zeno.

> In Plato's [[Parmenides]] (pp. 127, 128, Steph., pp. 6, 7, Bekk.) this dialectic is very well described, for Plato makes Socrates say of it: "Zeno in his writings asserts fundamentally the same as does Parmenides, that All is One, but he would feign delude us into believing that he was telling something new. Parmenides thus shows in his poems that All is One; Zeno, on the contrary, shows that the Many cannot be." Zeno replies, that "He wrote thus really against those who try to make Parmenides' position ridiculous, for they try to show what absurdities and self-contradictions can be derived from his statements; he thus combats those who deduce Being from the many, in order to show that far more absurdities arise from this than from the statements of Parmenides." That is the special aim of objective dialectic, in which we no longer maintain simple thought for itself, but see the battle fought with new vigour within the enemy's camp. Dialectic has in Zeno this negative side, but it bas also to be considered from its positive side.

> According to the ordinary ideas of science, where propositions result from proof, proof is the movement of intelligence, a connection brought about by mediation. Dialectic is either (a) external dialectic, in which this movement is different from the comprehension of the movement, or (b) not a movement of our intelligence only, but what proceeds from the nature of the thing itself, i.e. from the pure Notion of the content. The former is a manner of regarding. objects in such a way that reasons are revealed and new light thrown, by means of which all that was supposed to be firmly fixed, is made to totter; there may be reasons which are altogether external too, and we shall speak further of this dialectic when dealing with the Sophists. The other dialectic, however, is the immanent contemplation of the object; it is taken for itself, without previous hypothesis, idea or obligation, not under any outward conditions, laws or causes; we have to put ourselves right into the thing, to consider the object in itself, and to take it in the determinations which it has. In regarding it thus, it shows from itself that it contains opposed determinations, and thus breaks up; this dialectic we more especially find in the ancients. The subjective dialectic, which reasons from external grounds, is moderate, for it grants that: "In the right there is what is not right, and in the false the true." True dialectic leaves nothing whatever to its object, as if the latter were deficient on one side only; for it disintegrates itself in the entirety of its nature. The result of this dialectic is null, the negative; the affirmative in it does not yet appear. This true dialectic may be associated with the work of the Eleatics. But in their case the real meaning and quality of philosophic understanding was not great, for they got no further than the fact that through contradiction the object is a nothing.

> Zeno's dialectic of matter has not been refuted to the present day; even now we have not got beyond it, and the matter is left in uncertainty. Simplicius, writing on the Physics of [[Aristotle]] (p. 30), says: "Zeno proves that if the many is, it must be great and small; if great, the many must be infinite in number" (it must have gone beyond the manifold, as indifferent limit, into the infinite; but what is infinite is no longer large. and no longer many, for it is the negation of the many). "If small, it must be so small as to have no size," like atoms. "Here he shows that what has neither size, thickness nor mass, cannot be. For if it were added to another, it would not cause its increase; were it, that is to say, to have no size and be added thereto, it could not supplement the size of the other and consequently that which is added is nothing. Similarly were it taken away, the other would not be made less, and thus it is nothing. If what has being is, each existence necessarily has size and thickness, is outside of one another, and one is separate from the other; the same applies to all else (peri tou proucontoς), for it, too, has size, and in it there is what mutually differs (proexei autou ti). But it is the same thing to say something once and to say it over and over again; in it nothing can be a last, nor will there not be another to the other. Thus if many are, they are small and great; small, so that they have no size; great, so that they are infinite."

> Aristotle (Phys. VI. 9) explains this dialectic further; Zeno's treatment of motion was above all objectively dialectical. But the particulars which we find in the [[Parmenides]] of [[Plato]] are not his. For Zeno's consciousness we see simple unmoved thought disappear, but become thinking movement; in that he combats sensuous movement, he concedes it. The reason that dialectic first fell on movement is that the dialectic is itself this movement, or movement itself the dialectic of all that is. The thing, as self-moving, has its dialectic in itself, and movement is the becoming another, self-abrogation. If Aristotle says that Zeno denied movement because it contains an inner contradiction, it is not to be understood to mean that movement did. not exist at all. The point is not that there is movement and that this phenomenon exists; the fact that there is movement is as sensuously certain as that there are elephants; it is not in this sense that Zeno meant to deny movement. The point in question concerns its truth. Movement, however, is held to be untrue, because the conception of it involves a contradiction; by that he meant to say that no true Being eau be predicated of it.

> Zeno's utterances are to be looked at from this point of view, not as being directed against the reality of motion, as would at first appear, but as pointing out how movement must necessarily be determined, and showing the course which must be taken. Zeno now brings forward four different arguments against motion; the proofs rest on the infinite divisibility of space and time.

> (a) This is his first form of argument: — "Movement has no truth, because what is in motion must first reach the middle of the space before arriving at the end." Aristotle expresses this thus shortly, because he had earlier treated of and worked out the subject at length. This is to be taken as indicating generally that the continuity of space is pre-supposed. What moves itself must reach a certain, end, this way is a whole. In order to traverse the whole, what is in motion must first pass over the half, and now the end of this half is considered as being the end; but this half of space is again a whole, that which also has a half, and the half of this half must first have been reached, and so on into infinity. Zeno here arrives at the infinite divisibility of space; because space and time are absolutely continuous, there is no point at which the division can stop. Every dimension (and every time and space always have a dimension) is again divisible into two halves, which must be measured off; and however small a space we have, the same conditions reappear. Movement would be the act of passing through these infinite moments, and would therefore never end 4 thus what is in motion cannot reach its end. It is known how Diogenes of Sinope, the Cynic, quite simply refuted these arguments against movement; without speaking he rose and walked about, contradicting them by action. But when reasons are disputed, the only valid refutation is one derived from reasons; men have not merely to satisfy themselves by sensuous assurance, but also to understand. To refute objections is to prove their non-existence, as when they are made to fall away and can hence be adduced no longer; but it is necessary to think of motion as Zeno thought of it, and yet to carry this theory of motion further still.

> We have here the spurious infinite or pure appearance, whose simple principle Philosophy demonstrates as universal Notion, for the first time making its appearance as developed in its contradiction; in the history of Philosophy a consciousness of this contradiction is also attained. Movement, this pure phenomenon, appears as something thought and shown forth in its real being — that is, in its distinction of pure self-identity and pure negativity, the point as distinguished from continuity. To us there is no contradiction in the idea that the here of space and the now of time are considered as a continuity and length; but their Notion is self-contradictory. Self-identity or continuity is absolute cohesion, the destruction of all difference, of all negation, of being for self; the point, on the contrary, is pure being-for-self, absolute self-distinction and the destruction of all identity and all connection with what is different. Both of these, however, are, in space and time, placed in one; space and time are thus the contradiction; it is necessary, first of all, to show the contradiction in movement, for in movement that which is opposed is, to ordinary conceptions, inevitably manifested. Movement is just the reality of time and space, and because this appears and is made manifest, the apparent contradiction is demonstrated, a and it is this contradiction that Zeno notices. The limitation of bisection which is involved in the continuity of space, is not absolute limitation, for that which is limited is again continuity; however, this continuity is again not absolute, for the opposite has to be exhibited in it, the limitation of bisection; but the limitation of continuity is still not thereby established, the half is still continuous, and so on into infinity. In that we say "into infinity," we place before ourselves a beyond, outside of the ordinary conception, which cannot reach so far. It is certainly an endless going forth, but in the Notion it is present, it is a progression from one opposed determination to others, from continuity to negativity, from negativity to continuity; but both of these are before us. Of these moments one in the process may be called the true one; Zeno first asserts continuous progression in such a way that no limited space can be arrived at as ultimate, or Zeno upholds progression in this limitation.

> The general explanation which Aristotle gives to this contradiction, is that space and time are not infinitely divided, but are only divisible. But it now appears that, because they are divisible — that is, in potentiality — they must actually be infinitely divided, for else they could not be divided into infinity. That is the general answer of the ordinary man in endeavouring to refute the explanation of Aristotle. Bayle (Tom. IV. art. Zénon, not. E.) hence says of Aristotle's answer that it is "pitoyable: C'est se moquer du monde que de se servir de cette doctrine; car si Ia matière est divisible à l'infini, elle contient un nombre infini de parties. Ce n'est done point un infini en puissance, c'est un infini, qui existe réellement, actuellement. Mais quandmême on accorderait cet infini en puissance, qui deviendrait un infini par Ia division actuelle de ses parties, on ne perdrait pas ses avantages; car le mouvement est une chose, qui a la même vertu, que la division. Il touche une partie de l'espace sans toucher l'autre, et il les touche toutes les unes après les autres. N'est-ce pas les distinguer actuellement? N'est-ce pas faire ce que ferait un géomètre sur une table en tirant des lignes, qui désignassent tous les demiponces? II ne brise pas Ia table en demi-pouces, mais il y fait néanmoins une division, qui marque Ia distinction actuelle des parties; et ie ne crois pas qu'Aristote ent voulu nier, que si l'on tirait une infinité de lignes sur un pouce de matière, on n'y introduisit une division, qui reduirait en infini actuel ce qui n'etait selon lui qu'un infini virtuel." This si is good! Divisibility is, as potentiality, the universal; there is continuity as well as negativity or the point posited in it — but posited as moment, and not as existent in and for itself. I can divide matter into infinitude, but I only can do so; I do not really divide it into infinitude. This is the infinite, that no one of its moments has reality. It never does happen that, in itself, one or other — that absolute limitation or absolute continuity — actually comes into existence in such a way that the other moment disappears. There are two absolute opposites, but they are moments, i.e. in the simple Notion or in the universal, in thought, if you will; for in thought, in ordinary conception, what is set forth both is and is not at the same time. What is represented either as such. or as an image of the conception, is not a thing; it has no Being, and yet it is not nothing.

> Space and time furthermore, as quantum, form a limited extension, and thus can be measured off; just as I do not actually divide space, neither does the body which is in motion. The partition of space as divided, is not absolute discontinuity [Punktualität], nor is pure continuity the undivided and indivisible; likewise time is not pure negativity or discontinuity, but also continuity. Both are manifested in motion, in which the Notions have their reality for ordinary conception — pure negativity as time, continuity as space. Motion itself is just this actual unity in the opposition, and the sequence of both moments in this unity. To comprehend motion is to express its essence in the form of Notion, i.e., as unity of negativity and continuity; but in them neither continuity nor discreteness can be exhibited as the true existence. If we represent space or time to ourselves as infinitely divided, we have an infinitude of points, but continuity is present therein as a space which comprehends them; as Notion, however, continuity is the fact that all these are alike, and thus in reality they do not appear one out of the other like points. But both these moments make their appearance as existent; if they are manifested indifferently, their Notion is no longer posited, but their existence. In.them as existent, negativity is a limited size, and they exist as limited space and time; actual motion is progression through a limited space — and a limited time and not through infinite space and infinite time.

> That what is in motion must reach the half is the assertion of continuity, i.e. the possibility of division as mere possibility; it is thus always possible in every space, however small. It is said that it is plain that the half must be reached, but in so saying, everything is allowed, including the fact that it never will be reached; for to say so in one case, is the same as saying it an infinite number of times. We mean, on the contrary, that in a larger space the half can be allowed, but we conceive that we must somewhere attain to a space so small that no halving is possible, or an indivisible, non-continuous space which is no space. This, however, is false, for continuity is a necessary determination; there is undoubtedly a smallest in space, i.e. a negation of continuity, but the negation is something quite abstract. Abstract adherence to the subdivision indicated, that is, to continuous bisection into infinitude, is likewise false, for in the conception of a half, the interruption of continuity is involved. We must say that there is no half of space, for space is continuous; a piece of wood may be broken into two halves, but not space, and space only exists in movement. It might equally be said that space consists of an endless number of points, i.e. of infinitely many limits and thus cannot be traversed. Men think themselves able to go from one indivisible point to another, but they do not thereby get any further, for of these there is an unlimited number. Continuity is split up into its opposite, a number which is indefinite; that is to say, if continuity is not admitted, there is no motion. It is false to assert that it is possible when one is reached, or that which is not continuous; for motion is connection. Thus when it was said that continuity is the presupposed possibility of infinite division, continuity is only the hypothesis; but what is exhibited in this continuity is the being of infinitely many, abstractly absolute limits.

> (b) The second proof, which is also the presupposition of continuity and the manifestation of division, is called "Achilles, the Swift." The ancients loved to clothe difficulties in sensuous representations. Of two bodies moving in one direction, one of which is in front and the other following at a fixed distance and moving quicker than the first, we know that the second will overtake the first. But Zeno says, "The slower can never be overtaken by the quicker." And he proves it thus: "The second one requires a certain space of time to reach the place from which the one pursued started at the beginning of the given period." Thus during the time in which the second reached the point where the first was, the latter went over a new space which the second has again to pass through in a part of this period; and in this way it goes into infinity.

         c       d    e   f   g
         B       A

> B, for instance, traverses two miles (c d) in an hour, A in the same time, one mile (d e); if they are two miles (c d) removed from one another, B has in one hour come to where A was at the beginning of the hour. While B, in the next half hour, goes over the distance crossed by A of one mile (d e), A has got half a mile (e f) further, and so on into infinity. Quicker motion does not help the second body at all in passing over the interval of space by which he is behind: the time which he requires, the slower body always has at its avail in order to accomplish some, although an ever shorter advance; and this, because of the continual division, never quite disappears.

> Aristotle, in speaking of this, puts it shortly thus: "This proof asserts the same endless divisibility, but it is untrue, for the quick will overtake the slow body if the limits to be traversed be granted to it." This answer is correct and contains all that can be said; that is, there are in this representation two periods of time and two distances, which are separated from one another, i.e. they are limited in relation to one another; when, on the contrary, we admit that time and space are continuous, so that two periods of time or points of space are related to one another as continuous, they are, while being two, not two, but identical. In ordinary language we solve the matter in the easiest, way, for we say: "Because the second is quicker, it covers a greater distance in the same time as the slow; it can therefore come to the place from which the first started and get further still." After B, at the end of the first hour, arrives at d and A at e, A in one and the same period, that is, in the second hour, goes over the distance e g, and B the distance d g. But this period of time which should be one, is divisible into that in which B accomplishes d e and that in which B passes through e g. A has a start of the first, by which it gets over the distance e f, so that A is at f at the same period as B is at e. The limitation which, according to Aristotle, is to be overcome, which must be penetrated, is thus that of time; since it is continuous, it must, for the solution of the difficulty, be said that what is divisible into two spaces of time is to be conceived of as one, in which B gets from d to e and from e to g, while A passes over the distance c g. In motion two periods, as well as two points in space, are indeed one.

> If we wish to make motion clear to ourselves, we say that,,the body is in one place and then it goes to another; because it moves, it is no longer in the first, but yet not in the second; were it in either it would be at rest. Where then is it? If we say that it is between both, this is to convey nothing at all, for were it between both, it would be in a place, and this presents the same difficulty. But movement means to be in this place and not to be in it, and thus to be in both alike; this is the continuity of space and time which first makes motion possible. Zeno, in the deduction made by him, brought both these points into forcible opposition. The discretion of space and time we also uphold, but there must also be granted to them the over-stepping of limits, i.e. the exhibition of limits as not being, or as being divided periods of time, which are also not divided. In our ordinary ideas we find the same determinations as those on which the dialectic of Zeno rests; we arrive at saying, though unwillingly, that in one period two distances of space are traversed, but we do not say that the quicker comprehends two moments of time in one; for that we fix a definite space. But in order that the slower may lose its precedence, it must be said that it loses its advantage of a moment of time, and indirectly the moment of space.

> Zeno makes limit. division, the moment of discretion in space and time, the only element which is enforced in the whole of his conclusions, and hence results the contradiction. The difficulty is to overcome thought. for what makes the difficulty is always thought alone, since it keeps apart the moments of an object which in their separation are really united. It brought about the Fall, for man ate of the tree of the knowledge of good and evil; but it also remedies these evils.

> (c) The third form, according to Aristotle, is as follows: — Zeno says. "I The flying arrow rests, and for the reason that what is in motion is always in the self-same Now and the self-same Here, in the indistinguishable;" it is here and here and here. It can be said of the arrow that it is always the same, for it is always in the same space and the same time; it does not get beyond its space. does not take in another, that is, a greater or smaller space. That, however, is what we call rest and not motion. In the Here and Now, the becoming "other" is abrogated, limitation indeed being established, but only as moment; since in the Here and Now as such, there is no difference, continuity is here made to prevail against the mere belief in diversity. Each place is a different place, and thus the same; true, objective difference does not come forth in these sensuous relations, but in the spiritual.

> This is also apparent in mechanics; of two bodies the question as to which moves presents itself before us. It requires more than two places — three at least — to determine which of them moves. But it is correct to say this, that motion is plainly relative; whether in absolute space the eye, for instance, rests, or whether it moves, is all the same. Or, according to a proposition brought forward by Newton, if two bodies move round. one another in a circle, it may be asked whether the one rests or both move. Newton tries to decide this by means of an external circumstance, the strain on the string. When I walk on a ship in a direction opposed to the motion of the ship, this is in relation to the ship, motion, and in relation to all else, rest.

> In both the first proofs, continuity in progression has the predominance; there is no absolute limit, but an overstepping of all limits. Here the opposite is established; absolute limitation, the interruption of continuity, without however passing into something else; while discretion is pre-supposed, continuity is maintained. Aristotle says of this proof: "It arises from the fact that it is taken for granted that time consists of the Now; for if this is not conceded, the conclusions will not follow."

> (d) "The fourth proof," Aristotle continues, "is derived from similar bodies which move in opposite directions in the space beside a similar body, and with equal velocity, one from one end of the space, the other from the middle. It necessarily results from this that half the time is equal to the double of it. The fallacy rests in this, that Zeno supposes that what is beside the moving body, and what is beside the body at rest, move through an equal distance in equal time with equal velocity, which, however, is untrue."

                       1
                  E|-|-|-|-|F
             k    i   m
             C|-|-|-|-|D
             g n h
        A|-|-|-|-|B

> In a definite space such as a table (A B) let us suppose two bodies of equal length with it and with one another, one of which (C D) lies with one end (C) on the middle (g) of the table, and the other (B F), being in the same direction, has the point (B) only touching the end of the table (h); and supposing they move in opposite directions, and the former (C D) reaches in an hour the end (h) of the table; we have the result ensuing that the one (E F) passes in the half of the time through the same space (1 k) which the other does in the double (g h); hence the half is equal to the double. That is to say, this second passes (let us say, in the point 1) by the whole of the first C D. In the first half-hour 1 goes from m to i, while k only goes from g to n.

                       1
                  E|-|-|-|-|F
                   k o i   m
                  C|-|-|-|-|D
                   g n h
              A|-|-|-|-|B

>In the second half-hour 1 goes past o to k, and altogether passes from m to k, or the double of the distance.

                       1
                  E|-|-|-|-|F
                       k  o i   m
                       C|-|-|-|-|D
                   g n h
               A|-|-|-|-|B

> This fourth form deals with the contradiction presented in opposite motion; that which is common is given entirely to one body, while it only does part for itself. Here the distance travelled by one body is the sum of the distance travelled by both, just as when I go two feet east, and from the same point another goes two feet west, we are four feet removed from one another; in the distance moved both are positive, and hence have to be added together. Or if I have gone two feet forwards and two feet backwards, although I have walked four feet, I have not moved from the spot; the motion is then nil, for by going forwards and backwards an opposition ensues which annuls itself.

> This is the dialectic of Zeno; he had a knowledge of the determinations which our ideas of space and time contain, and showed in them their contradiction; Kant's antinomies do no more than Zeno did here. The general result of the Eleatic dialectic has thus become, "the truth is the one, all else is untrue," just as the Kantian philosophy resulted in "we know appearances only." On the whole the principle is the same; "the content of knowledge is only an appearance and not truth," but there is also a great difference present. That is to say, Zeno and the Eleatics in their proposition signified "that the sensuous world, with its multitudinous forms, is in itself appearance only, and has no truth." But Kant does not mean this, for he asserts: "Because we apply the activity of our thought to the outer world, we constitute it appearance; what is without, first becomes an untruth by the fact that we put therein a mass of determinations. Only our knowledge, the spiritual, is thus appearance; the world is in itself absolute truth; it is our action alone that ruins it, our work is good for nothing." It shows excessive humility of mind to believe that knowledge has no value; but Christ says, "Are ye not better than the sparrows?" and we are so inasmuch as we are thinking; as sensuous we are as good or as bad as sparrows. Zeno's dialectic has greater objectivity than this modern dialectic.

> Zeno's dialectic is limited to Metaphysics; later, with the Sophists, it became general. We here leave the Eleatic school. which perpetuates itself in Leucippus and, on the other side, in the Sophists, in such a way that these last extended the Eleatic conceptions to all reality, and gave to it the relation of consciousness; the former, however, as one who later on worked out the Notion in its abstraction, makes a physical application of it, and one which is opposed to consciousness. There are several other Eleatics mentioned, to Tennemann's surprise, who, however, cannot interest us. "It is so unexpected," he says (Vol. I., p. 190), "that the Eleatic system should find disciples; and yet Sextus mentions a certain Xeniades."


Zeno's paradox of motion

One of the kind of [[paradoxes]] considered by [[Zeno]]. Widely regarded as resolved by the modern concept of [[differentiation]] via [[limit of a sequence]] ([[analysis]], [[differential calculus]]) (see e.g. [Boyer 49, p. 267](#Boyer49)).

## References

* Wikipedia, _[Zeno's paradoxes -- Paradox of motion](http://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Paradoxes_of_motion)_

* Stanford Encyclopedia of Philosophy, _[Zeno's paradoxes --  The paradoxes of motion](http://plato.stanford.edu/entries/paradox-zeno/#ParMot)_

* {#Boyer49} Carl Benjamin Boyer, _The history of the Calculus and its conceptual development_, Dover 1949

* [[Georg Hegel]], chapter on _[[cohesion]]_ and _[[elasticity]]_ in _[[Encyclopedia of the Philosophical Sciences]]_, see also at _[Science of Logic --  Kohäsion](#Science+of+Logic#Kohaesion)_.

[[!redirects Zeno's paradox of motion]]
[[!redirects Zeno's paradoxes of motion]]

[[!redirects paradox of motion]]
[[!redirects paradoxes of motion]]


Zentralblatt MATH

[Zentralblatt MATH](https://zbmath.org/) (zbMATH) is an abstracting and reviewing service in [[mathematics]].

The editorial work is done by [[FIZ Karlsruhe]], a nonprofit organization.

## See also

* [[Mathematical Reviews]] (alias [[MathSciNet]])

* [[Math-Net.Ru]]

[[!redirects zbMATH]]


Zeph Landau

* [personal page](https://people.eecs.berkeley.edu/~landau/)

## Selected writings

On [[adiabatic quantum computation]]:

* [[Dorit Aharonov]], [[Wim van Dam]], [[Julia Kempe]], [[Zeph Landau]], [[Seth Lloyd]], [[Oded Regev]], *Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation*, SIAM Journal of Computing **37** 1 (2007) 166-194 [[arXiv:quant-ph/0405098](https://arxiv.org/abs/quant-ph/0405098), [jstor:20454175](https://www.jstor.org/stable/20454175), [doi:10.1109/FOCS.2004.8](https://doi.org/10.1109/FOCS.2004.8), [doi:10.1137/080734479](https://doi.org/10.1137/080734479)]

category: people


zero

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## Definition

The additive [[neutral element]] in the [[natural numbers]], [[integers]], [[real numbers]] and [[complex numbers]] is called **zero** and written as $0$.

More generally, in any [[abelian group]], or even any [[commutative monoid]], the group operation is often called 'addition' and written as $+$, and then the [[neutral element]] is called **zero** and written as $0$.  As a consequence, in any [[ring]], or more generally any [[rig]], the two binary operations are called 'multiplication' and 'addition', and the identity for addition is called **zero**.

## Categorification

[[categorification|Categorifying]] this idea, in any [[2-rig]] the additive identity may be called **zero**.  This is especially true in the case of a [[distributive category]], that is a category with (at least finitary) [[products]] and [[coproducts]], the former distributing over the latter.  In this case the [[initial object]], which serves as the identity for coproducts, is often called **zero**:

$$x + 0 \cong x$$

For example, in the category [[Set]], the [[empty set]] is often written $0$ in the category-theoretic literature.

In an [[abelian category]], the initial object is also [[terminal object|terminal]], and denoted $0$.  More generally, any object with this property is called a [[zero object]].

Categorifying [[horizontal categorification|horizontally]] instead, we get the notion of [[zero morphism]].

All these ideas can be, and have been, categorified further.

## Related concepts

* [[zero morphism]]

* [[trivial group]]

* [[one]]

* [[two]]

[[!redirects zero]]
[[!redirects 0]]


zero bundle

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## Idea

For any notion of [[bundles]] with [[fibers]] from a [[pointed category]], a *zero bundle* is a [[bundle]] all whose fibers are [[zero objects]].

The construction of zero-bundles typically (such as in the following examples) constitutes a [[bireflective subcategory]] inclusion

$$
  Spaces \hookrightarrow Bundles
$$

of the category of base spaces into the given category of bundles.

For example:

* in the context of [[vector bundles]] the zero-bundle over a base space $B$ is the bundle $B \times \{0\} \xrightarrow{pr_1} X$ all whose [[fibers]] are the [[zero]]-[[dimension of a vector space|dimensional]] [[vector space]], and the induced [[bireflective subcategory]] inclusion is that of base spaces into the corresponding category [[VectBund]];

* in the context of [[retractive spaces]] the zero-bundle over a base space $B$ is the [[identity]] map on $B$, all whose fibers are (equal or, in [[homotopy theory]], [[weak homotopy equivalence|equivalent]] to) the [[point]] $\ast$ regarded as a [[pointed space]];

* in the context of [[parameterized spectra]], the zero-bundle over a base space $X$ has all fibers the [[zero]]-[[spectrum]] $0_\bullet$ (i.e. the spectrum all whose components are [[contractible homotopy type|contractible]], $0_n \simeq \ast$ for all $n$).

(Beware that a zero-bundle is generally -- such as in the above examples -- *not* an [[empty bundle]].)

## Related concepts

* [[zero object]]

* [[bireflective subcategory]]

* [[empty bundle]]

[[!redirects zero bundles]]

[[!redirects zero-bundle]]
[[!redirects zero-bundles]]

[[!redirects zero vector bundle]]
[[!redirects zero vector bundles]]

[[!redirects zero-vector bundle]]
[[!redirects zero-vector bundles]]

[[!redirects zero spectrum bundle]]
[[!redirects zero spectrum bundles]]

[[!redirects zero-spectrum bundle]]
[[!redirects zero-spectrum bundles]]


zero function

A __zero function__ is a [[constant function]] whose constant value is [[zero]].

Formally, if $X$ is a [[set]] and $Y$ is a [[pointed set]] with [[basepoint]] $0$, then the __zero function__ from $X$ to $Y$ is
$$ 0\colon X \to Y\colon t \mapsto 0 .$$

If $X$ is also pointed, then the zero function must preserve basepoints, making this a case of a [[zero morphism]].  However, $X$ need not be pointed for the zero function to make sense.

[[!redirects zero function]]
[[!redirects zero functions]]


zero ideal

The __zero ideal__ or __trivial ideal__ of a [[ring]] $R$ is the two-sided [[ideal]] that consists entirely of the [[zero]] element.  It may be denoted $\{0\}$, $\mathbf{0}$, or simply $0$ (since it is the zero element of the [[rig of ideals]]).

We may generalize to a [[rig]], including the special case of a [[distributive lattice]] (in which the zero element is the [[bottom element]]), then generalize further to any [[poset]] with a bottom element $\bot$, in which the trivial ideal is $\{\bot\}$.

The trivial ideal of $R$ is the [[intersection]] of all of the ideals of $R$.  (If $R$ is a poset *without* a bottom element, then we may still consider the intersection of all of its ideals, but I\'m not sure if this deserves the name.)

[[!redirects zero ideal]]
[[!redirects zero ideals]]

[[!redirects trivial ideal]]
[[!redirects trivial ideals]]


zero locus

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## Idea

The _zero locus_ or _vanishing locus_ of a [[function]] is the set of points where it vanishes, in that it takes the value [[zero]].

## Definition

For $f \colon X \to \mathbb{A}$ a [[function]], its _zero locus_ is the [[preimage]] $f^{-1}(0)$ of [[zero]], hence the [[level set]] at 0.

## Properties

### Nullstellensatz

Hilbert's [[Nullstellensatz]] (German: "zero locus theorem") characterizes joint zero loci of [[ideals]] of functions in a [[polynomial ring]].

## Related concepts

* [[quadratic formula]]

* [[critical locus]], [[derived critical locus]]

* [[moment map]], [[Hamiltonian]]

* [[zero-set structure]]

[[!redirects zero loci]]
[[!redirects zero set]]
[[!redirects zero sets]]

[[!redirects vanishing locus]]
[[!redirects vanishing loci]]


zero morphism

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## Definition

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###### Definition

In a [[category]] $C$ with [[zero object]] $0$ the **zero morphism ** $0_{c,d} : c \to d$ between two [[object]]s $c, d \in C$ is the unique [[morphism]] that factors through $0$:

$$
  0_{c,d} : c \to 0 \to d
  \,.
$$

More generally, in any category [[enriched category|enriched]] over the [[closed monoidal category|closed monoidal]] [[category of pointed sets]] (with [[tensor product]] the [[smash product]]), the **zero morphism** $0_{c,d} : c \to d$ is the basepoint of the [[hom-object]] $[c,d]$.

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###### Remark

In fact, an enrichment over pointed sets consists precisely of the choice of a 'zero' morphism $0_{c,d}:c\to d$ for each pair of objects, with the property that $0_{c,d} \circ f = 0_{b,d}$ and $f\circ 0_{a,b} = 0_{a,c}$ for any morphism $f:b\to c$.  Such an enrichment is unique if it exists, for if we are given a different collection of zero morphisms $0'_{c,d}$, we must have
$$0'_{c,d} = 0'_{c,d} \circ 0_{c,c} = 0_{c,d}$$
for any $c,d$.  Thus, the existence of zero morphisms can be regarded as a [[stuff, structure, property|property]] of a category, rather than structure on it.  (To be more precise, it is an instance of [[property-like structure]], since not every functor between categories with zero morphisms will necessarily preserve the zero morphisms, although an [[equivalence of categories]] will.)

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## Examples

See at _[[zero object]]_ for examples.

## Related concepts

* [[zero]]

* [[zero function]]

* [[null homotopy]]

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[[!redirects zero map]]
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