topology

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###Context###
#### Topology
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[[!include topology - contents]]
=--
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> This page is about topology as a **field of [[mathematics]]**.  For topology as a **[[structured set|structure]]** on a [[set]], see _[[topological space]]_.

> Parts of this page exists also in a German language version, see at _[[Topologie]]_.

***

#Contents#
* table of contents
{:toc}

\linebreak

\linebreak

> I believe that we lack another analysis properly geometric or linear  which expresses location directly as algebra expresses magnitude.

> [[G. W. Leibniz]] (letter to Huygens 1679, according to [Bredon 93, p. 430](#Bredon93))

## Idea

__Topology__ is one of the basic fields of [[mathematics]].  The term is also used for a particular structure in a [[topological space]]; see [[topological structure]] for that.

The subject of topology deals with the expressions of [[continuous map|continuity]] and [[boundary]], and studying the geometric properties of (originally: [[metric space|metric]]) [[spaces]] and relations of subspaces, which do not change under continuous deformations, regardless to other (such as in their metric properties).

Topology as a structure enables one to model [[continuity]] and [[convergence]] locally. More recently, in metric spaces, topologists and geometric group theorists started looking at asymptotic properties at large, which are in some sense dual to the standard topological structure and are usually referred to as _coarse topology_.

There are many cousins of the concept of [[topological spaces]], e.g. [[sites]], [[locales]], [[topoi]], [[higher topoi]], [[uniformity spaces]] and so on, which specialize or generalize some aspect or structure usually found in [[Top]].

One of the tools of topology, [[homotopy theory]], has long since crossed the boundaries of topology and applies to many other areas, thanks to many examples and motivations as well as of abstract categorical frameworks for homotopy like Quillen [[model categories]], Brown's [[category of fibrant objects|categories of fibrant objects]] and so on.

## Introduction
 {#Introduction}

The following gives a quick introduction to some of the core concepts and tools of topology:

* _[Continuity](#Continuity)_

* _[Topological spaces](#TopologicalSpaces)_

* _[Homeomorphism](#Homeomorphisms)_

* _[Homotopy](#Homotopy)_

* _[Connected components](#ConnectedComponents)_

* _[Fundamental group](#FundamentalGroups)_

* _[Covering spaces](#CoveringSpaces)_

A detailed introduction is going to be at _[[Introduction to Topology]]_.

### Continuity
 {#Continuity}

The key idea of topology is to study [[spaces]] with "[[continuous maps]]" between them. The concept of continuity was made precise first in [[analysis]], in terms of [[epsilontic analysis]] of [[open balls]], recalled as def. \ref{EpsilonDeltaDefinitionOfContinuity} below. Then it was realized that this has a more elegant formulation in terms of the more general concept of _[[open sets]]_, this is prop. \ref{ContinuityBetweenMetricSpacesInTermsOfOpenSets} below. Adopting the latter as the definition leads to the concept of [[topological spaces]], def. \ref{TopologicalSpace} below.

First recall the basic concepts from [[analysis]]:

+-- {: .num_defn #MetricSpace}
###### Definition
**(metric space)**

A _[[metric space]]_ is

1. a [[set]] $X$ (the "underlying set");

1. a [[function]] $d \;\colon\; X \times X \to [0,\infty)$ (the "distance function") from the [[Cartesian product]] of the set with itself to the [[nonnegative number|non-negative]] [[real numbers]]

such that for all $x,y,z \in X$:

1. $d(x,y) = 0 \;\;\Leftrightarrow\;\; x = y$

1. (symmetry) $d(x,y) = d(y,x)$

1. ([[triangle inequality]]) $d(x,y)+ d(y,z) \geq d(x,z)$.

=--

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

Every [[normed vector space]] $(V, {\vert - \vert})$
becomes a [[metric space]] according to def. \ref{MetricSpace} by setting

$$
  d(x,y) \coloneqq {\vert x-y \vert}
  \,.
$$

=--

+-- {: .num_defn #EpsilonDeltaDefinitionOfContinuity}
###### Definition
**(epsilontic definition of continuity)**

For $(X,d_X)$ and $(Y,d_Y)$ two [[metric spaces]] (def. \ref{MetricSpace}), then a [[function]]

$$
  f \;\colon\; X \longrightarrow Y
$$

is said to be _continuous at a point $x \in X$_ if for every $\epsilon \gt 0$ there exists $\delta\gt 0$ such that

$$
  d_X(x,y) \lt \delta \;\;\Rightarrow\;\; d_Y(f(x), f(y)) \lt \epsilon
$$

or equivalently such that

$$
  f(\;B_x^\circ(\delta)\;) \;\subset\; B^\circ_{f(x)}(\epsilon)
$$

where $B^\circ$ denotes the [[open ball]] (definition \ref{OpenBalls}).

The function $f$ is called just _continuous_ if it is continuous at every point $x \in X$.

=--

We now reformulate this analytic concept in terms of the simple but important concept of _[[open sets]]_:

+-- {: .num_defn #OpenBalls}
###### Definition
**(open ball)**

Let $(X,d)$, be a [[metric space]]. Then for every element $x \in X$ and every  $\epsilon \in \mathbb{R}_+$ a [[positive number|positive]] [[real number]], write

$$
  B^\circ_x(\epsilon)
    \;\coloneqq\;
  \left\{
    y \in X \;\vert\; d(x,y) \lt \epsilon
  \right\}
$$

for the [[open ball]] of [[radius]] $\epsilon$ around $x$.

=--

+-- {: .num_defn #OpenSubsetsOfAMetricSpace}
###### Definition
**(neighbourhood and open set)**

Let $(X,d)$ be a [[metric space]] (def. \ref{MetricSpace}). Say that

1. A _[[neighbourhood]]_ of a point $x \in X$ is a [[subset]] $x \in U \subset X$ which contains some [[open ball]] $B_x^\circ(\epsilon)$ around $x$ (def. \ref{OpenBalls}).

1. An _[[open subset]]_ of $X$ is a [[subset]] $U \subset X$ such that for every $x \in U$ it also contains a [[neighbourhood]] of $x$.

=--

The following picture shows a point $x$, some [[open balls]] $B_i$ containing it, and two of its [[neighbourhoods]] $U_i$:

> graphics grabbed from [Munkres 75](#Munkres75)

+-- {: .num_prop #ContinuityBetweenMetricSpacesInTermsOfOpenSets}
###### Proposition
**(rephrasing continuity in terms of open sets)**

A [[function]] $f \colon X \to Y$ between [[metric spaces]] (def. \ref{MetricSpace}) is continuous in the [[epsilontic analysis|epsilontic]] sense of def. \ref{EpsilonDeltaDefinitionOfContinuity} precisely if it has the property that its [[pre-images]] of [[open subsets]] of $Y$ (in the sense of def. \ref{OpenSubsetsOfAMetricSpace}) are open subsets of $X$.

=--

+-- {: principle}
**principle of continuity**

$\,$ $\,$ _Pre-Images of open subsets are open._

=--

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

First assume that $f$ is continuous in the epsilontic sense. Then for $O_Y \subset Y$ any [[open subset]] and $x \in f^{-1}(O_Y)$ any point in the pre-image, we need to show that there exists a [[neighbourhood]] of $x$ in $f^{-1}(O_Y)$. But by assumption there exists an [[open ball]] $B_x^\circ(\epsilon)$ with $f(B_x^\circ(\epsilon)) \subset O_Y$. Since this is true for all $x$, by definition this means that $f^{-1}(O_Y)$ is open in $X$.

Conversely, assume that $f^{-1}$ takes open subsets to open subsets. Then for every $x \in X$ and $B_{f(x)}^\circ(\epsilon)$ an [[open ball]] around its image, we need to produce an open ball $B_x^\circ(\delta)$ in its pre-image. But by assumption $f^{-1}(B_{f(x)}^\circ(\epsilon))$ contains a [[neighbourhood]] of $x$ which by definition means that it contains such an open ball around $x$.

=--

### Topological spaces
 {#TopologicalSpaces}

Therefore we should pay attention to [[open subsets]]. It turns out that the following closure
property is what _characterizes_ the concept:

+-- {: .num_prop }
###### Proposition
**(closure properties of open sets in a metric space)**

The collection of [[open subsets]] of a [[metric space]] $(X,d)$ as in def. \ref{OpenSubsetsOfAMetricSpace} has the following properties:

1. The [[intersection]] of any [[finite number]] of open subsets is again an open subset.

1. The [[union]] of any [[set]] of open subsets is again an open subset.

In particular

* the [[empty set]] is open (being the union of no subsets)

and

* the whole set $X$ itself is open (being the intersection of no subsets).

=--

This motivates the following generalized definition:

+-- {: .num_defn #TopologicalSpace}
###### Definition
**(topological spaces)**

Given a [[set]] $X$, then a _topology_ on $X$ is a collection $\tau$ of [[subsets]] of $X$ called the _[[open subsets]]_, hence a [[subset]] of the [[power set]]

$$
  \tau \subset P(X)
$$

such that this is closed under forming

1. finite [[intersections]];

1. arbitrary [[unions]].

A _[[topological space]]_ is a set $X$ equipped with such a [[topology]].

=--

The following shows all the topologies on the 3-element set (up to permutation of elements)

> graphics grabbed from [Munkres 75](#Munkres75)

It is now immediate to formally implement the

+-- {: principle}
**principle of continuity**

$\,$ $\,$ _Pre-Images of open subsets are open._

=--

+-- {: .num_defn #ContinuousMaps}
###### Definition
**(continuous maps)**

A _[[continuous function]]_ between [[topological spaces]]

$$
  f \colon (X, \tau_X) \to (Y, \tau_Y)
$$

is a [[function]] between the underlying sets,

$$
  f \colon X \longrightarrow Y
$$

such that [[pre-images]] under $f$ of open subsets of $Y$ are open subsets of $X$.

=--

The simple definition of [[open subsets]] and the simple _principle of continuity_ gives topology its fundamental and universal flavor. The combinatorial nature of these definitions makes
topology closely related to [[formal logic]] (for more on this see at _[[locale]]_).

\begin{remark}
**(the category of topological spaces)**
\linebreak
The [[composition]] of [[continuous functions]] is clearly [[associativity|associative]] and [[unitality|unital]].

One says that

1. [[topological spaces]] constitute the [[objects]]

1. [[continuous maps]] constitute the [[morphisms]] ([[homomorphisms]])

of  a _[[category]]_. The _[[category of topological spaces]]_ ("[[Top]]" for short).

It is useful to depict collections of [[objects]] with [[morphisms]] between them
by [[diagrams]], like this one:

> graphics grabbed from [Lawvere-Schanuel 09](#category+theory#LawvereSchanuel09).

\end{remark}

Our motivating example now reads:

+-- {: .num_example #MetricTopology}
###### Example
**(metric topology)**

Let $(X,d)$ be a [[metric space]]. Then the collection of open subsets in def. \ref{OpenSubsetsOfAMetricSpace} constitutes a _[[topological space|topology]]_ on the set $X$, making it a _[[topological space]]_ in the sense of def. \ref{TopologicalSpace}. This is called the _[[metric topology]]_.

Stated more concisely: the [[open balls]] in a metric space constitute a "[[basis of a topology|basis]]" for the [[metric topology]].

=--

One point of the general definition of "[[topological space]]" is that it admits constructions which intuitively should exist on "continuous spaces", but which do not in general exist, for instance, as metric spaces:

+-- {: .num_example #DiscreteTopologicalSpace}
###### Example
**(discrete topological space)**

For $S$ a bare [[set]], then the _[[discrete topology]]_ on $S$ regards _every_ subset of $S$ as an [[open subset]].

=--

+-- {: .num_example #SubspaceTopology}
###### Example
**(subspace topology)**

Let $(X, \tau_X)$ be a [[topological space]], and let $X_0 \hookrightarrow X$ be a [[subset]] of the underlying set. Then the corresponding _[[topological subspace]]_ has $X_0$ as its underlying set, and its open subsets are those subsets of $X_0$ which arise as restrictions of open subsets of $X$.

(Also called the _[[initial topology]]_ of the inclusion map.)

The picture on the right shows two open subsets inside the [[square]], regarded as a [[topological subspace]] of the [[plane]] $\mathbb{R}^2$:

> graphics grabbed from [Munkres 75](#Munkres75)

=--

+-- {: .num_example #QuotientTopologicalSpace}
###### Example
**(quotient topological space)**

Let $(X,\tau_X)$ be a [[topological space]] (def. \ref{TopologicalSpace}) and let

$$
  R_\sim \subset X \times X
$$

be an [[equivalence relation]] on its underlying set. Then the _[[quotient topological space]]_  has

* as underlying set the [[quotient set]] $X_{\sim}$, hence the set of [[equivalence classes]],

and

* a subset $O \subset X_{\sim}$ is declared to be an [[open subset]] precisely if its [[preimage]] $\pi^{-1}(O)$ under the canonical [[projection map]]

  $$
    \pi \colon X \to X_\sim
  $$

  is open in $X$.

(This is also called the _[[final topology]]_ of the projection $\pi$.)

=--

The above picture shows on the left the [[square]] (a [[topological subspace]] of the [[plane]]),
then in the middle the resulting [[quotient topological space]] obtained by
identifying two opposite sides (the _[[cylinder]]_), and on the right the further quotient obtained by
identifying the remaining sides (the _[[torus]]_).

> graphics grabbed from [Munkres 75](#Munkres75)

+-- {: .num_example #ProductTopologicalSpace}
###### Example
**(product topological space)**

For $X$ and $Y$ two [[topological spaces]], then the [[product topological space]] $X \times Y$
has

* as underlying set the [[Cartesian product]] of the underlying sets of $X$ and $Y$,

and

* its [[open sets]] are those subsets $O \subset X \times Y$ of the Cartesian product such that for all point $(x,y) \in O$
there exists open sets $x \in O_x \subset X$ and $y \in O_Y \subset Y$ such that $O_x \times O_y \subset O$.

> graphics grabbed from [Munkres 75](#Munkres75)

=--

These constructions of [[discrete topological spaces]], [[quotient topological spaces]], [[topological subspaces]] and of [[product topological spaces]] are simple examples of **[[limits]]** and of **[[colimits]]** of topological spaces. The [[category]] [[Top]] of topological spaces has the convenient property that _all_ [[limits]] and [[colimits]] (over [[small diagrams]]) exist in it. (For more on this see at _[Top -- Universal constructions](Top#UniversalConstructions)_.)

### Homeomorphism
 {#Homeomorphisms}

With the [[objects]] ([[topological spaces]]) and the [[morphisms]] ([[continuous maps]]) of the [[category]] [[Top]] of topology thus defined, we obtain the concept of "sameness" in topology.

To make this precise, one says that a [[morphism]]

$$
  X \overset{f}{\to} Y
$$

in a [[category]] is an _[[isomorphism]]_ if there exists a morphism going the other way around

$$
  X \overset{f^{-1}}{\longleftarrow} Y
$$

which is an [[inverse]] in the sense that

$$
  f \circ f^{-1} = id_Y \;\;\;\;\; and \;\;\;\;\; f^{-1} \circ f = id_X
  \,.
$$

+-- {: .num_defn #Homeomorphism}
###### Definition
**(homeomorphisms)**

An [[isomorphism]] in the [[category]] [[Top]] of [[topological spaces]] with [[continuous functions]] between them is called a _[[homeomorphism]]_.

Hence this is a [[continuous function]]

$$
  f \;\colon\; X \longrightarrow Y
$$

such that there exists an [[inverse]] [[morphism]], namely a [[continuous function]] the other way around

$$
  X \longleftarrow Y \;\colon\; f^{-1}
$$

such that

$$
  f \circ f^{-1} = id_{Y} \;\;\;and\;\;\; f^{-1} \circ f = id_{X}
  \,.
$$

=--

> graphics grabbed from [Munkres 75](#Munkres75)

+-- {: .num_example #OpenBallsHomeomorphicToRn}
###### Example
**(open interval homeomorphic to the real line)**

The open [[interval]] $(-1,1)$ is [[homeomorphic]] to all of the [[real line]]

$$
  (-1,1) \underset{homeo}{\simeq} \mathbb{R}^1
  \,.
$$

An [[inverse]] pair of [[continuous functions]] is for instance given by

$$
  \array{
    f &\colon&  \mathbb{R}^1 &\longrightarrow& (-1,+1)
    \\
    && x &\mapsto& \frac{x}{\sqrt{1+ x^2}}
  }
$$

and

$$
  \array{
    f^{-1} &\colon&  (-1,+1) &\longrightarrow& \mathbb{R}^1
    \\
    && x &\mapsto& \frac{x}{\sqrt{1 - x^2}}
  }
  \,.
$$

Generally, every [[open ball]] in $\mathbb{R}^n$ (def. \ref{OpenBalls}) is [[homeomorphic]] to all of $\mathbb{R}^n$.

=--

+-- {: .num_example #HomeomorphismBetweenTopologicalAndCombinatorialCircle}
###### Example
**(interval glued at endpoints is homeomorphic to the circle)**

As topological spaces, the [[interval]] with its two endpoints identified is [[homeomorphic]] (def. \ref{Homeomorphism}) to the standard [[circle]]:

$$
  [0,1]_{/(0 \sim 1)} \;\; \underset{homeo}{\simeq} \;\; S^1
  \,.
$$

More in detail: let

$$
  S^1 \hookrightarrow \mathbb{R}^2
$$

be the unit [[circle]] in the [[plane]]

$$
  S^1 = \{(x,y) \in \mathbb{R}^2, x^2 + y^2 = 1\}
$$

equipped with the [[subspace topology]] (example \ref{SubspaceTopology}) of the plane $\mathbb{R}^2$,
which itself equipped with its standard [[metric topology]] (example \ref{MetricTopology}).

Moreover, let

$$
  [0,1]_{/(0 \sim 1)}
$$

be the [[quotient topological space]] (example \ref{QuotientTopologicalSpace}) obtained from the [[interval]] $[0,1] \subset \mathbb{R}^1$ with its [[subspace topology]] by applying the [[equivalence relation]] which identifies the two endpoints (and nothing else).

Consider then the function

$$
  f \;\colon\; [0,1] \longrightarrow S^1
$$

given by

$$
  t \mapsto (cos(2\pi t), sin(2\pi t))
  \,.
$$

This has the property that $f(0) = f(1)$, so that it descends to the [[quotient topological space]]

$$
  \array{
    [0,1] &\overset{}{\longrightarrow}& [0,1]_{/(0 \sim 1)}
    \\
    & {}_{\mathllap{f}}\searrow & \downarrow^{\mathrlap{\tilde f}}
    \\
    && S^1
  }
  \,.
$$

We claim that $\tilde f$ is a [[homeomorphism]] (definition \ref{Homeomorphism}).

First of all it is immediate that $\tilde f$ is a [[continuous function]]. This follows immediately from the fact that $f$ is a [[continuous function]] and by definition of the [[quotient topology]] (example \ref{QuotientTopologicalSpace}).

So we need to check that $\tilde f$ has a continuous inverse function. Clearly the restriction of $f$ itself to the open interval $(0,1)$ has a continuous inverse. It fails to have a continuous inverse on $[0,1)$ and on $(0,1]$ and fails to have an inverse at all on [0,1], due to the fact that $f(0) = f(1)$. But the relation quotiented out in $[0,1]_{/(0 \sim 1)}$ is exactly such as to fix this failure.

=--

Similarly:

The [[square]] $[0,1]^2$ with two of its sides identified is the [[cylinder]], and with also the other two sides
identified is the [[torus]]:

If the sides are identified with opposite orientation, the result is the _[[Möbius strip]]_:

> graphics grabbed from [Lawson 03](#Lawson03)

$\,$

Important examples of pairs of spaces that are _not_ homeomorphic include the following:

\begin{theorem}\label{TopologicalInvarianceOfDimension}
**([[topological invariance of dimension]])**
\linebreak
For $n_1, n_2 \in \mathbb{N}$ but $n_1 \neq n_2$, then the [[Cartesian spaces]] $\mathbb{R}^{n_1}$ and $\mathbb{R}^{n_2}$ are _not_ [[homeomorphic]].

More generally, an [[open set]] in $\mathbb{R}^{n_1}$ is never homeomorphic to an open set in $\mathbb{R}^{n_2}$ if $n_1 \neq n_2$.
\end{theorem}

The proof of theorem \ref{TopologicalInvarianceOfDimension} is surprisingly hard, given how obvious the statement seems intuitively. It requires tools from a field called _[[algebraic topology]]_ (notably [[Brouwer's fixed point theorem]]).

We showcase some basic tools of [[algebraic topology]] now and demonstrate the nature of their usage by proving two very simple special cases of the [[topological invariance of dimension]] (prop. \ref{TopologicalInvarianceOfDimensionFirstSimpleCase} and prop. \ref{topologicalInvarianceOfDimensionSecondSimpleCase} below).

+-- {: .num_example }
###### Example
**(homeomorphism classes of surfaces)**

The [[2-sphere]] $S^2 = \{(x,y,z) \in \mathbb{R}^3 \vert x^2 + y^2 + z^2 = 1\}$ is _not_ [[homeomorphic]] to the [[torus]] $T^2 = S^1 \times S^1$.

Generally the [[homeomorphism]] class of a [[closed manifold|closed]] [[orientable]] [[surface]] is determined by the number of "holes" it has, its _[[genus of a surface|genus]]_.

=--

$\,$

### Homotopy
  {#Homotopy}

We have seen above that for $n \geq 1$  then the [[open ball]] $B_0^\circ(1)$ in $\mathbb{R}^n$ is _not_ [[homeomorphic]] to, notably, the point $\ast = \mathbb{R}^0$ (example \ref{OpenBallsHomeomorphicToRn}, theorem \ref{TopologicalInvarianceOfDimension}). Nevertheless, intuitively the $n$-ball is a "continuous deformation" of the point, obtained as the radius of the $n$-ball tends to zero.

This intuition is made precise by observing that there is a [[continuous function]] out of the [[product topological space]]
(example \ref{ProductTopologicalSpace}) of the open ball with the closed [[interval]]

$$
  \eta \colon [0,1] \times B_0^\circ(1)  \longrightarrow \mathbb{R}^n
$$

which is given by rescaling:

$$
  (t,x) \mapsto t \cdot x
  \,.
$$

This continuously interpolates between the open ball and the point in that for $t = 1$ then it restricts to the defining inclusion $B_0^\circ(1)$, while for $t = 0$ then it restricts to the map constant on the origin.

We may summarize this situation by saying that there is a
[[diagram]] of [[continuous functions]] of the form

$$
  \array{
    B_0^\circ(1) \times \{0\}
    \\
    \downarrow & \searrow^{\mathrlap{x \mapsto 0}}
    \\
    [0,1] \times B_0^\circ(1)
    &\overset{(t,x) \mapsto t \cdot x}{\longrightarrow}&
    \mathbb{R}^n
    \\
    \uparrow & \nearrow_{\mathrlap{inclusion}}
    \\
    B_0^\circ(1) \times \{1\}
  }
$$

Such "continuous deformations" are called _[[homotopies]]_:

+-- {: .num_defn #LeftHomotopy}
###### Definition
**(homotopy)**

For $f,g\colon X \longrightarrow Y$ two [[continuous functions]] between [[topological spaces]] $X,Y$, then a _[[left homotopy|(left) homotopy]]_

$$
  \eta \colon f \,\Rightarrow_L\, g
$$

is a [[continuous function]]

$$
  \eta \;\colon\; X \times I \longrightarrow Y
$$

out of the [[product topological space]] (example \ref{ProductTopologicalSpace})
of the open ball with the standard interval, such that this fits into a [[commuting diagram]] of the form

$$
  \array{
     {0} \times X
     \\
     {}^{\mathllap{(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{f}}
     \\
     [0,1] \times X  &\stackrel{\eta}{\longrightarrow}& Y
     \\
     {}^{\mathllap{(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{g}}
     \\
     \{1\} \times X
  }
  \,.
$$

> graphics grabbed from J. Tauber [here](http://jtauber.com/blog/2005/07/01/path_homotopy/)

=--

+-- {: .num_defn #HomotopyEquivalence}
###### Definition
**(homotopy equivalence)**

A [[continuous function]] $f \;\colon\; X \longrightarrow Y$
is called a _[[homotopy equivalence]]_ if

1. there exists a continuous function the other way around,
$g \;\colon\; Y \longrightarrow X$, and

1. [[left homotopies]], def. \ref{LeftHomotopy}, from the two composites to the identity:

  $$
    \eta_1 \;\colon\; f\circ g \Rightarrow_L id_Y
  $$

  and

  $$
    \eta_2 \;\colon\; g\circ f \Rightarrow_L id_X
    \,.

  $$

=--

+-- {: .num_example}
###### Example
**(open ball is contractible)**

Any [[open ball]] (or closed ball), hence (by example \ref{OpenBallsHomeomorphicToRn}) any [[Cartesian space]]
is [[homotopy equivalence|homotopy equivalent]] to the point

$$
  \mathbb{R}^n \underset{homotopy}{\simeq} \ast
  \,.
$$

=--

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

The following three [[graphs]]

(i.e. the evident [[topological subspaces]] of the [[plane]] $\mathbb{R}^2$ that these pictures indicate) are not [[homeomorphic]]. But they are [[homotopy equivalence|homotopy equivalent]], in fact they are each homotopy equivalent to the [[disk]] with two points removed, by the homotopies indicated by the following pictures:

> graphics grabbed from [Hatcher](homotopy+equivalence#Hatcher)

=--

$\,$

### Connected components
 {#ConnectedComponents}

Using the concept of [[homotopy]] one obtains the basic tool of [[algebraic topology]], namely the construction of algebraic [[homotopy invariants]] of topological spaces. We introduce the simplest and indicate their use.

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

A [[homotopy]] between two points

$x,y \;\colon\; \ast \to X$

is a continuous [[path]] between these points.

=--

+-- {: .num_defn #pi0}
###### Definition
**(connected components)**

The set of [[homotopy classes]] of points in a [[topological space]] $X$ is called its set of
_path-[[connected components]]_, denoted.

$$
  \pi_0(X) \in Set
  \,.
$$

If $\pi_0(X) \simeq \ast$ consists of a single element, we say that $X$ is a _[[path-connected topological space]]_.

If $f \colon X \to Y$ is a [[continuous map]] between [[topological spaces]], then on [[homotopy classes]] of points it induces a [[function]] between the corresponding connected components, which we denote by:

$$
  \pi_0(f) \;\colon\; \pi_0(X) \longrightarrow \pi_0(Y)
  \,.
$$

=--

This construction is evidently compatible with [[composition]], in that

$$
  \pi_0(g \circ f) = \pi_0(g) \circ \pi_0(f)
$$

and it evidently is [[unitality|unital]], in that

$$
  \pi_0(id_X) = id_{\pi_{0}(X)}
  \,.
$$

One summarizes this by saying that $\pi_0$ is a _[[functor]]_ from the [[category]] [[Top]] of [[topological spaces]] to the [[category]] [[Set]] of [[sets]], denoted

$$
  \pi_0 \;\colon\; Top \longrightarrow Set
  \,.
$$

An immediate but important consequence is this:

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

If two [[topological spaces]] have sets of [[connected components]] that are not in [[bijection]], then those spaces are not [[homeomorphic]] to each other:

$$
  \pi_0(X) \neq \pi_0(Y)
  \;\;\;
    \Rightarrow
  \;\;\,
  X \underset{homeo}{\neq} Y
$$

=--

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

Since $\pi_0$ is [[functor|functorial]], it immediately follows that it sends [[isomorphisms]] to [[isomorphisms]], hence [[homeomorphisms]] to [[bijections]]:

$$
  \begin{aligned}
     & f \circ g = id \;\;and\;\; g \circ f = id
    \\
    \Rightarrow \;\;\;\;\;\;&
    \pi_0(f \circ g) = \pi_0(id) \;\;and \;\; \pi_0(g \circ f) = \pi_0(id)
    \\
    \Leftrightarrow \;\;\;\;\;\;
    &
    \pi_0(f) \circ \pi_0(g) = id \;\;and \;\; \pi_0(g) \circ \pi_0(f) = id
  \end{aligned}
  \,.
$$

=--

This means that we may use path connected components as a first "topological invariant" that allows us to distinguish some topological spaces.

+-- {: principle}
**principle of algebraic topology**

$\,\,$ _Use topological invariants to distinguish topological spaces._

=--

As an example for how this is being used, we have the following proof of a simple special case of the [[topological invariance of dimension]] (theorem \ref{TopologicalInvarianceOfDimension}):

+-- {: .num_prop #TopologicalInvarianceOfDimensionFirstSimpleCase}
###### Proposition
**([[topological invariance of dimension]] -- first simple case)**

The [[Cartesian spaces]] $\mathbb{R}^1$ and $\mathbb{R}^2$ are not [[homeomorphic]] (def. \ref{Homeomorphism}).

=--

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

Assume there were a [[homeomorphism]]

$$
  f \colon \mathbb{R}^1 \longrightarrow \mathbb{R}^2
$$

we will derive a contradiction. If $f$ is a homeomorphism, then clearly so is its restriction to the [[topological subspaces]] (example \ref{SubspaceTopology}) obtained by removing $0 \in \mathbb{R}^1$ and $f(0) \in \mathbb{R}^2$.

$$
  f
    \;\colon\;
  (\mathbb{R}^1-\{0\})
    \longrightarrow
  (\mathbb{R}^2 - \{f(0)\})
  \,.
$$

It follows that we would get a bijection of [[connected components]] between $\pi_0(\mathbb{R}^1 - \{0\})$ and $\pi_0(\mathbb{R}^2  - \{f(0)\})$. But clearly the first set has two elements, while the second has just one:

$$
  \pi_0(\mathbb{R}^1-\{0\})
    \;\neq\;
  \pi_0(\mathbb{R}^2 - \{f(0)\})
  \,.
$$

=--

The key lesson of the proof of prop. \ref{TopologicalInvarianceOfDimensionFirstSimpleCase} is its strategy:

+-- {: principle}
**principle of algebraic topology**

$\,\,$ _Use topological invariants to distinguish topological spaces._

=--

Of course in practice one uses more sophisticated invariants than just $\pi_0$.

The next topological invariant after the [[connected components]] is the _[[fundamental group]]:

### Fundamental group
 {#FundamentalGroups}

+-- {: .num_defn #FundamentalGroup}
###### Definition
**(fundamental group)**

Let $X$ be a [[topological space]] and let $x \in X$ be a chosen point. Then write

$$
  \pi_1(X,x)
    \;\in\;
  Grp
$$

for, to start with, the set of [[homotopy classes]] of [[paths]] in $X$ that start and end at $x$. Such paths are also called the continuous [[loops]] in $X$ based at $x$.

1. Under concatenation of loops, $\pi_1(X,x)$ becomes a [[semi-group]].

1. The constant loop is a [[neutral element]] under this composition (thus making $\pi_1(X,x)$ a "[[monoid]]").

1. The  reverse of a loop is its [[inverse]] in $\pi_1(X,x)$, making $\pi_1(X,x)$ indeed into a [[group]].

This is called the _[[fundamental group]]_ of $X$ at $x$.

=--

The following picture indicates the four non-equivalent non-trivial generators of the [[fundamental group]] of the
oriented [[surface]] of [[genus of a surface|genus]] 2:

> graphics grabbed from [Lawson 03](#Lawson03)

Again, this operation is [[functor|functorial]], now on the [[category]] $Top^{\ast/}$
of [[pointed topological spaces]], whose [[objects]] are
topological spaces equipped with a chosen point, and whose [[morphisms]] are [[continuous maps]]
$f \colon X \to Y$ that take the chosen basepoint of $X$ to that of $Y$:

$$
  \pi_1 \;\colon\; Top^{\ast/} \longrightarrow Grp
  \,.
$$

As $\pi_0$, so also $\pi_1$ is a topological invariant. As before, we may use this to prove a simple case of the theorem of the [[topological invariance of dimension]]:

+-- {: .num_defn #SimplyConnected}
###### Definition

A topological space $X$ for which

1. $\pi_0(X) \simeq \ast$ ([[path-connected topological space|path connected]], def. \ref{pi0})

1. $\pi_1(X,x) \simeq 1$ (the [[fundamental group]] is [[trivial group|trivial]], def. \ref{FundamentalGroup}),

is called _[[simply connected topological space|simply connected]]_.

=--

+-- {: .num_prop #topologicalInvarianceOfDimensionSecondSimpleCase}
###### Proposition
**([[topological invariance of dimension]] -- second simple case)**

There is _no_ [[homeomorphism]] between $\mathbb{R}^2$ and $\mathbb{R}^3$.

=--

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

Assume there were such a homeomorphism $f$; we will derive a contradiction.

If $f$ is a homeomorphism, then so is its restriction to removing the origin from $\mathbb{R}^2$
and $f(0)$ from $\mathbb{R}^3$:

$$
  (\mathbb{R}^2 - \{0\})
    \longrightarrow
  (\mathbb{R}^3 - \{f(0)\})
  \,.
$$

Thse two spaces are both path-connected, hence $\pi_0$ does not distiguish them.

But they do have different [[fundamental groups]] $\pi_1$:

1. The fundamental group of $\mathbb{R}^{2} - \{0\}$ is $\mathbb{Z}$ (counting the winding of loops around the removed point).
We discuss this further below in example \ref{FundamentalGroupOfTheCircle}.

1. The fundamental group of $\mathbb{R}^3 - \{f(0)\}$ is trivial: because the single removed point is
no obstruction to sliding loops past it and contracting them.

But since passing to fundamental groups is [[functor|functorial]], the same argument
as in the proof of prop. \ref{ConnectedComponentsDistinctImpliesHeomeClassesDistinct}
shows that $f$ cannot be an [[isomorphism]], hence not a [[homeomorphism]].

=--

We now discuss a "dual incarnation" of fundamental groups, which often helps to compute them.

### Covering spaces
 {#CoveringSpaces}

+-- {: .num_defn #CoveringSpace}
###### Definition
**(covering space)**

A _[[covering space]]_ of a [[topological space]] $X$ is a [[continuous map]]

$$
  p \colon E \to X
$$

such that there exists an [[open cover]] $\underset{i}{\sqcup}U_i \to X$, such that restricted
to each $U_i$ then $E \to X$ is [[homeomorphic]] over $U_i$
to the [[product topological space]] (example \ref{ProductTopologicalSpace}) of $U_i$
with the [[discrete topological space]] (example \ref{DiscreteTopologicalSpace}) on a [[set]] $F_i$

$$
  \array{
    \underset{i}{\sqcup} U_i \times F_i &\longrightarrow&  E
    \\
    \downarrow &(pb)& \downarrow^{\mathrlap{p}}
    \\
    \underset{i}{\sqcup} U_i &\underset{}{\longrightarrow}& X
  }
  \,.
$$

For $x \in U_i \subset X$ a point, then the elements in $F_x  = F_i$ are called the _leaves_ of the covering at $x$.

=--

+-- {: .num_example #kForlCovringOfCircle}
###### Example
**(covering of circle by circle)**

Regard the [[circle]] $S^1 = \{ z \in \mathbb{C}  \;\vert\; {\vert z\vert} = 1 \}$ as the [[topological subspace]]
of elements of unit [[absolute value]] in the [[complex plane]].
For $k \in \mathbb{N}$, consider the continuous function

$$
  p \coloneqq (-)^k \;\colon\; S^1 \longrightarrow S^1
$$

given by taking a complex number to its $k$th power. This may be thought of as the
result of "winding the circle $k$ times around itself".
Precisely, for $k \geq 1$ this is a [[covering space]]
(def. \ref{CoveringSpace}) with $k$ leaves at each point.

> graphics grabbed from [Hatcher](homotopy+equivalence#Hatcher)

=--

+-- {: .num_example #CoveringOfCircleByRealLine}
###### Example
**(covering of circle by real line)**

Consider the [[continuous function]]

$$
  \exp(2 \pi i(-)) \;\colon\; \mathbb{R}^1 \longrightarrow S^1
$$

from the [[real line]] to the [[circle]],
which,

1. with the circle regarded as the unit circle in the [[complex plane]] $\mathbb{C}$, is given by

   $$
     t \mapsto \exp(2\pi i t)
   $$

1. with the circle regarded as the unit circle in $\mathbb{R}^2$, is given by

   $$
     t \mapsto ( cos(2\pi t), sin(2\pi t) )
     \,.
   $$

We may think of this as the result of "winding the line around the circle ad infinitum".
Precisely, this is a [[covering space]] (def. \ref{CoveringSpace}) with the [[leaves]] at each point forming the
set $\mathbb{Z}$ of [[natural numbers]].

=--

+-- {: .num_defn #ActionOfFundamentalGroupOnFibersOfCovering}
###### Definition
**(action of fundamental group on fibers of covering)**

Let $E \overset{\pi}{\longrightarrow} X$ be a [[covering space]] (def. \ref{CoveringSpace})

Then for $x \in X$ any point, and any choice of element $e \in F_x$ of the [[leaf space]] over $x$,
there is, up to [[homotopy]], a unique way to lift a representative
path in $X$ of an element $\gamma$ of the the [[fundamental group]] $\pi_1(X,x)$ (def. \ref{FundamentalGroup}) to a
continuous path in $E$ that starts at $e$. This path necessarily ends at some (other) point $\rho_\gamma(e) \in F_x$
in the same [[fiber]]. This construction provides a [[function]]

$$
  \array{
    \rho &\colon& F_x \times \pi_1(X,x) &\longrightarrow& F_x
    \\
    && (e,\gamma) &\mapsto& \rho_\gamma(e)
  }
$$

from the [[Cartesian product]] of the [[leaf space]] with the [[fundamental group]]. This function
is compatible with the [[group]]-structure on $\pi_1(X,x)$, in that the following [[commuting diagram|diagrams commute]]:

$$
  \array{
    F_x \times \{const_x\} && \longrightarrow && F_x \times \pi_1(X,x)
    \\
    & {}_{\mathllap{id}}\searrow && \swarrow_{\mathrlap{\rho}}
    \\
    && F_x
  }
  \;\;\;\;\;\;
  \left(
  \array{
    \text{the neutral element,}
    \\
    \text{i.e. the constant loop,}
    \\
    \text{acts trivially}
  }
  \right)
$$

and

$$
  \array{
    F_x \times \pi_1(X,x) \times \pi_1(X,x)
      &\overset{\rho \times id}{\longrightarrow}&
    F_x \times \pi_1(X,x)
    \\
    {}^{\mathllap{id \times ((-)\cdot(-))}}\downarrow && \downarrow^{\mathrlap{\rho}}
    \\
    F_x \times \pi_1(X,x)
      &\underset{\rho}{\longrightarrow}&
    F_x
  }
  \;\;\;\;\;\;
  \left(
  \array{
    \text{acting with two group elements }
    \\
    \text{is the same as}
    \\
    \text{first multiplying them}
    \\
    \text{ and then acting with their product element}
  }
  \right)
  \,.
$$

One says that $\rho$ is an _[[action]]_ or _[[permutation representation]]_ of $\pi_1(X,x)$ on $F_x$.

For $G$ any [[group]], then there is a [[category]] $G Set$ whose [[objects]] are [[sets]]
equipped with an [[action]] of $G$, and whose [[morphisms]] are [[function|functions]] which respect
these actions. The above construction yields a [[functor]]

$$
  Cov(X) \longrightarrow \pi_1(X,x) Set
  \,.
$$

=--

+-- {: .num_example }
###### Example
**(three-sheeted covers of the circle)**

There are, up to [[isomorphism]], three different 3-sheeted [[covering spaces]] of the [[circle]] $S^1$.

The one from example \ref{kForlCovringOfCircle} for $k = 3$. Another one. And the trivial one.
Their corresponding [[permutation actions]] may be seen from the pictures on the right.
> graphics grabbed from [Hatcher](homotopy+equivalence#Hatcher)

=--

We are now ready to state the main theorem about the [[fundamental group]]. Except that it
does require the following slightly technical condition on the base topological space.
This condition is satisfied for all "reasonable" topological spaces:

+-- {: .num_defn #SemiLocallySimplyConnected}
###### Definition
**(semi-locally simply connected)**

A [[topological space]] $X$ is called

1. _[[locally path-connected]]_ if for every point $x \in X$ and for every [[neighbourhood]] $x \in U \subset X$ there
   exists a neighbourhood $x \in V \subset U$ such that $V$ is [[path-connected topological space|path-connected]] (def. \ref{pi0});

1. _[[semi-locally simply connected topological space|semi-locally simply connected]]_ if every point $x \in X$ has a [[neighbourhood]] $x \in U \subset X$ such that the induced morphism of [[fundamental groups]] $\pi_1(U,x) \to \pi_1(X,x)$ is trivial (i.e. sends everything to the [[neutral element]]).

=--

+-- {: .num_theorem #FundamentalTheoremOfCoveringSpaces}
###### Theorem
**([[fundamental theorem of covering spaces]])**

Let $X$ be a [[topological space]] which is [[connected topological space|path-connected]] (def. \ref{pi0}),
[[locally path-connected topological space|locally path connected]] (def. \ref{SemiLocallySimplyConnected})
and [[semi-locally simply connected topological space|semi-locally simply connected]] (def. \ref{SemiLocallySimplyConnected}). Then
for any $x \in X$ the functor
$$
  Fib_x \;\colon\; Cov(X) \overset{}{\longrightarrow} \pi_1(X,x) Set
  \,.
$$
from def. \ref{ActionOfFundamentalGroupOnFibersOfCovering} that describes
the [[action]] of the [[fundamental group]] of $X$ on the set of [[leaves]] over $x$
has the following property:

1. every [[isomorphism class]] of $\pi_1(X,x)$-[[actions]] is in the image of the functor (one says: the functor is _[[essentially surjective functor|essentially surjective]]_);

1. for any two covering spaces $E_1, E_2$ of $X$ then the map on [[hom-sets|morphism sets]]

   $$
     Fib_x \;\colon\; Hom_{Cov(X)}(E_1, E_2) \longrightarrow Hom( Fib_x(E_1), Fib_x(E_2) )
   $$

   is a [[bijection]] (one says the functor is a _[[fully faithful functor]]_ ).

A functor with these two properties one calls an _[[equivalence of categories]]_:

$$
  Cov(X) \overset{\simeq}{\longrightarrow} \pi_1(X,x) Set
  \,.
$$

=--

This has some interesting implications:

Every sufficiently nice topological space $X$ as above has a covering which is [[simply connected topological space|simply connected]]
(def. \ref{SimplyConnected}). This is the covering corresponding, under the [[fundamental theorem of covering spaces]]
(theorem \ref{FundamentalTheoremOfCoveringSpaces}) to the action of $\pi_1(X)$ on itself.
This is called the _[[universal covering space]]_ $\hat X \to X$. The above theorem implies that the
[[fundamental group]] itself may be recovered as the [[automorphisms]] of the universal covering space:

$$
  \pi_1(X) \simeq Aut_{Cov_{/X}}(\hat X, \hat X)
  \,.
$$

+-- {: .num_example #FundamentalGroupOfTheCircle}
###### Example
**(computing the fundamental group of the circle)**

The covering $\exp(2\pi i(-)) \;\colon\; \mathbb{R}^1 \to S^1$ from example \ref{CoveringOfCircleByRealLine}
is [[simply connected topological space|simply connected]] (def. \ref{SimplyConnected}),  hence must be the [[universal covering space]], up to [[homeomorphism]].

It is fairly straightforward to see that the only [[homeomorphisms]] from $\mathbb{R}^1$ to itself
over $S^1$ are given by [[integer]] translations by $n \in \mathbb{N} \hookrightarrow \mathbb{R}$:

$$
  \array{
    \mathbb{R}^1
      &&
        \underoverset{\simeq}{t \mapsto t + n}{\longrightarrow}
      &&
    \mathbb{R}^1
    \\
    & {}_{\mathllap{\exp(2 \pi i(-))}}\searrow && \swarrow_{\mathrlap{\exp(2 \pi i(-))}}
    \\
    && S^1
  }
  \,.
$$

Hence

$$
  Aut_{Cov_{/S^1}}(\hat S^1, \hat S^1) \simeq \mathbb{Z}
$$

and hence the [[fundamental group]] of the [[circle]] is the additive group of [[integers]]:

$$
  \pi_1(S^1) \simeq \mathbb{Z}
  \,.
$$

=--

$\,$

***

$\,$

## Basic facts
 {#BasicFacts}

* [[Hausdorff spaces are sober]], [[schemes are sober]]

* [[CW-complexes are Hausdorff]]

* [[compact Hausdorff spaces are normal]]

* [[continuous image of a compact space is compact]]

* [[closed subspaces of compact Hausdorff spaces are equivalently compact subspaces]]

* [[quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff]]

## Central theorems
 {#CentralTheorems}

* [[Tietze extension theorem]]

* [[Tychonoff theorem]]

* [[Heine-Borel theorem]]

* [[topological invariance of dimension]]

* [[Brouwer's fixed point theorem]]

* [[Jordan curve theorem]]

## Related entries
 {#RelatedEntries}

* [[algebraic topology]]

* [[computational topology]]

* [[applied topology]]

  [[topological data analysis]]

The following is an (incomplete) list of available $n$Lab entries related to topology.

### Topological spaces

* [[Top]] [[CW-complex]], [[general topology]]
* [[induced topology]], [[subspace]], [[interior]], [[boundary]], [[closure]]

* [[product topological space]], [[disjoint union topological space]]

* [[sphere]], [[metric space]], [[metrizable space]]
* [[topological space]], [[continuous map]], [[homeomorphism]], [[neighborhood]]
* [[pointed space]], [[contractible space]], [[connected space]], [[second countable space]]
* [[convergence space]], [[pretopological space]], [[pseudotopological space]], [[coarse topology]]
* [[metric space]], [[filtered space]], [[connected filtered space]], [[complete space]], [[net]], [[Polish space]]
* [[separation axioms]], [[Hausdorff space]], [[regular space]], [[normal space]]
* [[noetherian topological space]], [[irreducible topological space]]
* [[compact space]], [[locally compact space]], [[compactum]], [[paracompact space]]
* [[Frechet-Uryson space]], [[sequential space]], [[uniform space]]
* [[convenient category of topological spaces]], [[compactly generated space]]
* [[nice topological space]], [[nice category of spaces]]
* [[pointless topology]], [[locale]], [[cover]], [[site]], [[ringed space]]
[[sphere]]
* [[Sierpinski space]], [[Warsaw circle]]

See also [[examples in topology]].

### Manifolds and generalizations

* [[manifold]], [[smooth manifold]], [[generalized smooth space]], [[diffeological space]], [[smooth space]], [[Froelicher space]]
* [[differential topology]], [[microbundle]], [[cobordism]], [[derived smooth manifold]]

* [[low-dimensional topology]], [[3-manifold]]

  [[quantum topology]]

### Algebraic topology and homotopy theory

* [[homotopy]], [[homotopy inverse]], [[homotopy theory]], [[homotopy equivalence]], [[rational homotopy theory]]

* [[shape theory]], [[algebraic topology]]

* [[model category]], [[category of fibrant objects]], [[Quillen adjunction]], [[Quillen equivalence]], [[category with weak equivalences]],

* [[model 2-category]], [[model stack]]

* [[Reedy category]], [[Reedy model structure]], [[generalized Reedy category]]
* [[model structure on chain complexes]], [[model structure on crossed complexes]]
* [[homotopy hypothesis]], [[homotopy theory of Grothendieck]], [[test category]]

* [[homotopy limit]], [[homotopy colimit]], [[homotopy pullback]], [[homotopy image]], [[homotopy coherent category theory]], [[homotopy coherent nerve]], [[Brown representability theorem]]
* [[homotopy type]], [[homotopy 1-type]], [[homotopy 2-type]], [[homotopy 3-type]], [[homotopy n-type]]
* [[deformation retract]], [[neighborhood retract]], [[Postnikov system]]
* [[fundamental group]], [[homotopy group]], [[suspension]], [[Freudenthal suspension theorem]], [[delooping]]
* [[loop space]], [[free loop space object]],
* [[cup product]], [[Dold-Thom theorem]]

#### Topological homotopy theory

* [[classical model structure on topological spaces]]

* [[topological group]], [[H-space]], [[co-H-space]]
* [[principal bundle]], [[fiber bundle]], [[vector bundle]], [[principal 2-bundle]], [[trivial bundle]]
* [[fibration]], [[Hurewicz fibration]], [[Hurewicz connection]], [[homotopy lifting property]], [[Dold fibration]]
* [[homotopy extension property]], [[Hurewicz cofibration]], [[deformation retract]], [[model structure on topological spaces]], [[Strøm's theorem]]
* [[cocylinder]], [[cylinder object]], [[path object]], [[mapping cone]], [[path groupoid]], [[path n-groupoid]], [[fundamental groupoid]]
* [[classifying space]], [[Eilenberg-MacLane space]],[[Moore space]], [[Moore path category]]
* [[spectrum]], [[smash product of spectra]], [[symmetric spectrum]], [[stable (infinity,1)-category]], [[commutative ring spectrum]]

#### Simplicial homotopy theory

* [[classical model structure on simplicial sets]]

* [[simplicial set]], [[simplex category]], [[simplicial identities]], [[simplicial object]], [[cosimplicial object]]
* [[simplicial complex]], [[singular simplicial complex]],
* [[boundary of a simplex]], [[simplicial skeleton]], [[category of simplices]]
* [[combinatorial spectrum]], [[simplicial groupoid]], [[simplicial model category]]
* [[simplicially enriched category]], [[quasicategory]], [[Segal category]], [[complete Segal space]]
* [[Kan fibration]], [[Kan complex]], [[nerve]], [[nerve and realization]], [[simplicial homotopy]], [[simplicial homotopy group]], [[simplicial local system]]
* [[simplicial presheaf]], [[local model structure on simplicial presheaves]], [[local model structure on simplicial sheaves]]
* [[marked simplicial set]], [[model structure on marked simplicial over-sets]], [[infinity topos]], [[Higher Topos Theory]]
* [[dendroidal set]], [[model structure on dendroidal sets]], [[cubical set]], [[cellular set]], [[cyclic object]], [[Theta space]]

### Sheaves, stacks, cohomology

* [[site]], [[sheaf]], [[flabby sheaf]], [[local epimorphism]], [[etale space]], [[hypercover]]
* [[topos]]
* [[cohomology]], [[sheaf cohomology]], [[abelian sheaf cohomology]], [[local system]], [[category of sheaves]]
* [[Čech cohomology]], [[Bredon cohomology]], [[twisted cohomology]], [[twisting cochain]]
* [[nonabelian algebraic topology]], [[nonabelian cocycle]], [[nonabelian cohomology]]
* [[principal infinity-bundle]], [[BrownAHT]], [[category of fibrant objects]]
* [[topological K-theory]], [[gerbe]], [[twisted K-theory]], [[Karoubi K-theory]], [[differential K-theory]], [[K-theory spectrum]]
* [[de Rham cohomology]], [[string topology]]
* [[Thom space]], [[Thom bundle]], [[fiber integration]], [[equivariant cohomology]]
* [[Poincaré duality]], [[Spanier-Whitehead duality]]
* [[generalized cohomology]], [[generalized (Eilenberg-Steenrod) cohomology]], [[generalized (Eilenberg-Steenrod) homotopy]]
* [[topological stack]], [[orbispace]]
* [[topological quantum field theory]], [[tmf]], [[cobordism hypothesis]], [[topological T-duality]]
* [[characteristic class]], [[Chern-Weil theory]]

### Non-commutative topology

* [[noncommutative topology]]

  * [[C-star-algebra]], [[KK-theory]]

### Topological physics

In [[solid state physics]]:

* [[topological phases of matter]]

  * [[topological insulators]], [[topological semi-metals]]

  * [[symmetry protected topological phases]]

  * [[topological order]]

* [[K-theory classification of topological phases of matter]]

In [[string theory]]:

* [[K-theory classification of D-brane charge|D-brane charge quantization in topological K-theory]]

### Computational Topology

* [[computational topology]]

* [[topological data analysis]]

* [[topological machine learning]]

* [[sources in computational topology]]

[[!include topology -- references]]


topology - contents

**[[topology]]** ([[point-set topology]], [[point-free topology]])

see also _[[differential topology]]_, _[[algebraic topology]]_, _[[functional analysis]]_ and _[[topological homotopy theory|topological]] [[homotopy theory]]_

[[Introduction to Topology|Introduction]]

**Basic concepts**

* [[open subset]], [[closed subset]], [[neighbourhood]]

* [[topological space]], [[locale]]

* [[base for the topology]], [[neighbourhood base]]

* [[finer topology|finer/coarser topology]]

* [[topological closure|closure]], [[topological interior|interior]], [[topological boundary|boundary]]

* [[separation axiom|separation]], [[sober topological space|sobriety]]

* [[continuous function]], [[homeomorphism]]

* [[uniformly continuous function]]

* [[topological embedding|embedding]]

* [[open map]], [[closed map]]

* [[sequence]], [[net]], [[sub-net]], [[filter]]

* [[convergence]]

* [[category]] [[Top]]

  * [[convenient category of topological spaces]]

**[Universal constructions](Top#UniversalConstructions)**

* [[initial topology]], [[final topology]]

* [[subspace]], [[quotient space]],

* fiber space, [[space attachment]]

* [[product space]], [[disjoint union space]]

* [[mapping cylinder]], [[mapping cocylinder]]

* [[mapping cone]], [[mapping cocone]]

* [[mapping telescope]]

* [[colimits of normal spaces]]

**[[stuff, structure, property|Extra stuff, structure, properties]]**

* [[nice topological space]]

* [[metric space]], [[metric topology]], [[metrisable space]]

* [[Kolmogorov space]], [[Hausdorff space]], [[regular space]], [[normal space]]

* [[sober space]]

* [[compact space]], [[proper map]]

  [[sequentially compact topological space|sequentially compact]], [[countably compact topological space|countably compact]], [[locally compact topological space|locally compact]], [[sigma-compact topological space|sigma-compact]], [[paracompact space|paracompact]], [[countably paracompact topological space|countably paracompact]], [[strongly compact topological space|strongly compact]]

* [[compactly generated space]]

* [[second-countable space]], [[first-countable space]]

* [[contractible space]], [[locally contractible space]]

* [[connected space]], [[locally connected space]]

* [[simply-connected space]], [[locally simply-connected space]]

* [[cell complex]], [[CW-complex]]

* [[pointed topological space|pointed space]]

* [[topological vector space]], [[Banach space]], [[Hilbert space]]

* [[topological group]]

* [[topological vector bundle]], [[topological K-theory]]

* [[topological manifold]]

**Examples**

* [[empty space]], [[point space]]

* [[discrete space]], [[codiscrete space]]

* [[Sierpinski space]]

* [[order topology]], [[specialization topology]], [[Scott topology]]

* [[Euclidean space]]

  * [[real line]], [[plane]]

*  [[cylinder]], [[cone]]

* [[sphere]], [[ball]]

* [[circle]], [[torus]], [[annulus]], [[Moebius strip]]

* [[polytope]], [[polyhedron]]

* [[projective space]] ([[real projective space|real]], [[complex projective space|complex]])

* [[classifying space]]

* [[configuration space (mathematics)|configuration space]]

* [[path]], [[loop]]

* [[mapping spaces]]: [[compact-open topology]], [[topology of uniform convergence]]

  * [[loop space]], [[path space]]

* [[Zariski topology]]

* [[Cantor space]], [[Mandelbrot space]]

* [[Peano curve]]

* [[line with two origins]], [[long line]], [[Sorgenfrey line]]

* [[K-topology]], [[Dowker space]]

* [[Warsaw circle]], [[Hawaiian earring space]]

**Basic statements**

* [[Hausdorff spaces are sober]]

* [[schemes are sober]]

* [[continuous images of compact spaces are compact]]

* [[closed subspaces of compact Hausdorff spaces are equivalently compact subspaces]]

* [[open subspaces of compact Hausdorff spaces are locally compact]]

* [[quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff]]

* [[compact spaces equivalently have converging subnet of every net]]

  * [[Lebesgue number lemma]]

  * [[sequentially compact metric spaces are equivalently compact metric spaces]]

  * [[compact spaces equivalently have converging subnet of every net]]

  * [[sequentially compact metric spaces are totally bounded]]

* [[continuous metric space valued function on compact metric space is uniformly continuous]]

* [[paracompact Hausdorff spaces are normal]]

* [[paracompact Hausdorff spaces equivalently admit subordinate partitions of unity]]

* [[closed injections are embeddings]]

* [[proper maps to locally compact spaces are closed]]

* [[injective proper maps to locally compact spaces are equivalently the closed embeddings]]

* [[locally compact and sigma-compact spaces are paracompact]]

* [[locally compact and second-countable spaces are sigma-compact]]

* [[second-countable regular spaces are paracompact]]

* [[CW-complexes are paracompact Hausdorff spaces]]

**Theorems**

* [[Urysohn's lemma]]

* [[Tietze extension theorem]]

* [[Tychonoff theorem]]

* [[tube lemma]]

* [[Michael's theorem]]

* [[Brouwer's fixed point theorem]]

* [[topological invariance of dimension]]

* [[Jordan curve theorem]]

**Analysis Theorems**

* [[Heine-Borel theorem]]

* [[intermediate value theorem]]

* [[extreme value theorem]]

**[[topological homotopy theory]]**

* [[left homotopy]], [[right homotopy]]

* [[homotopy equivalence]], [[deformation retract]]

* [[fundamental group]], [[covering space]]

* [[fundamental theorem of covering spaces]]

* [[homotopy group]]

* [[weak homotopy equivalence]]

* [[Whitehead's theorem]]

* [[Freudenthal suspension theorem]]

* [[nerve theorem]]

* [[homotopy extension property]], [[Hurewicz cofibration]]

* [[topological cofiber sequence|cofiber sequence]]

* [[Strøm model category]]

* [[classical model structure on topological spaces]]


topology - countability axioms

[[!redirects topology - global countability axioms]]

\begin{tikzcd}[column sep=normal, row sep=large]
	&
	&
	& \scriptsize \text{metrisable}
	& \substack{ \sigma\text{-locally finite } \\ \text{base}\wedge \text{regular} \wedge \mathrm{T}_2 }
	&
	\\
	& \text{second countable}
	&
	& \sigma\text{-locally discrete base}
	& \sigma\text{-locally finite base}
	& \text{first countable}
	\\
	  \text{separable}
	&
	& \text{Lindel\"of}
	& \substack{\text{weakly Lindel\"of }\wedge \\ \sigma\text{-locally finite base}}
	&
	& \text{Fr\'echet-Urysohn}
	\\
	&
	&
	&
	&
	& \text{sequential}
	\\
	  \text{countable chain condition}
	&
	& \text{weakly Lindel\"of}
	&
	&
	& \text{countably tight}
	\arrow[Rightarrow, from=3-4, to=2-2]
	\arrow[Rightarrow, from=3-4, to=5-3]
	\arrow[Rightarrow, from=3-4, to=2-5]
	\arrow[Rightarrow, from=2-2, to=3-1, "\text{CC}" description]
	\arrow[Rightarrow, from=2-2, to=3-3, "\text{CC}" description]
	\arrow[Rightarrow, from=3-1, to=3-3, "\text{if $X$ metacompact}"]
	\arrow[Rightarrow, from=3-3, to=5-3]
	\arrow[Rightarrow, from=3-1, to=5-3, "\text{AC}" {description, near end}]
	\arrow[Rightarrow, from=2-2, to=2-4]
	\arrow[Rightarrow, from=3-1, to=5-1]
	\arrow[Rightarrow, from=2-4, to=2-5]
	\arrow[Rightarrow, from=2-5, to=2-6]
	\arrow[Rightarrow, from=2-6, to=3-6]
	\arrow[Rightarrow, from=3-6, to=4-6]
	\arrow[Rightarrow, from=4-6, to=5-6]
	\arrow[Rightarrow, from=1-4, to=2-4]
	\arrow[Leftrightarrow, from=1-5, to=1-4, "\substack{\text{Nagata-Smirnov} \\ \text{metrization thm.} }" above]
	\arrow[Rightarrow, from=1-5, to=2-5]
	\arrow[Rightarrow, from=5-1, to=3-3, "\text{AC}" {description, near end}, "\text{if $X$ paracompact}" {near start, below right}]
\end{tikzcd}

Axioms: [[axiom of choice]] (AC), [[countable choice]] (CC).

### Properties ###

* [[second-countable space|second-countable]]: there is a [[countable set|countable]] [[topological base|base]] of the topology.

* [[metrisable topological space|metrisable]]: the topology is induced by a metric.

* $\sigma$-locally discrete base: the topology of $X$ is generated by a [[countably locally discrete set of subsets|$\sigma$-locally discrete]] [[topological base|base]].

* $\sigma$-locally finite base: the topology of $X$ is generated by a [[countably locally finite set of subsets|countably locally finite]] [[topological base|base]].

* [[separable space|separable]]: there is a countable [[dense subspace|dense]] subset.

* [[Lindelöf topological space|Lindelöf]]: every [[open cover]] has a [[countable cover|countable]] sub-cover.

* [[weakly Lindelöf topological space|weakly Lindelöf]]: every [[open cover]] has a [[countable set|countable]] subcollection the union of which is dense.

* [[metacompact space|metacompact]]: every open cover has a point-finite open refinement.

* [[countable chain condition]]: A family of pairwise disjoint open subsets is at most countable.

* [[first-countable space|first-countable]]: every point has a countable [[neighborhood base]]

* [[Frechet-Uryson space]]: the closure of a set $A$ consists precisely of all limit points of sequences in $A$

* [[sequential topological space]]: a set $A$ is closed if it contains all limit points of sequences in $A$

* [[countably tight space|countably tight]]: for each subset $A$ and each point $x\in \overline A$ there is a countable subset $D\subseteq A$ such that $x\in \overline D$.

### Implications ###

* [[open covers of metric spaces have open countably locally discrete refinements|a metric space has a $\sigma$-locally discrete base]]

* [[Nagata-Smirnov metrization theorem]]

* a second-countable space has a [[countably locally finite set of subsets|$\sigma$-locally finite]] [[topological base|base]]:  take the the collection of singeltons of all elements of a countable cover of $X$.

* second-countable spaces are separable: use the axiom of [[countable choice]] to choose a point in each set of a countable cover.

* [[second-countable spaces are Lindelöf]].

* [[weakly Lindelöf spaces with countably locally finite base are second countable]].

* [[separable metacompact spaces are Lindelöf]].

* separable spaces satisfy the countable chain condition: given a dense set $D$ and a family $\{U_\alpha : \alpha \in A\}$, the map $D \cap \bigcup_{\alpha \in A} U_\alpha \to A$ assigning $d$ to the unique $\alpha \in A$ with $d \in U_\alpha$ is surjective.

* separable spaces are weakly Lindelöf: given a countable dense subset and an open cover [[axiom of choice|choose]] for each point of the subset an open from the cover.

* Lindelöf spaces are trivially also weakly Lindelöf.

* a space with a $\sigma$-locally finite base is first countable: obviously, every point is contained in at most countably many sets of a $\sigma$-locally finite base.

* a first-countable space is obviously Fréchet-Urysohn.

* a Fréchet-Uryson space is obviously sequential.

* a sequential space is obviously countably tight.

* [[paracompact spaces satisfying the countable chain condition are Lindelöf]].


topology -- references

## References
 {#References}

### Historical origins
 {#ReferencesHistoricalOrigins}

The general idea of [[topology]] goes back to:

* [[Henri Poincaré]], *[[Analysis Situs]]*, Journal de l'École Polytechnique **2** 1 (1895) 1–123 \[gallica:12148/bpt6k4337198/f7, Engl: [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf), [[Stillwell_AnalysisSitus.pdf:file]]\]

The notion of [[topological space]] involving [[neighbourhoods]] was first developed, for the special case now known as *[[Hausdorff spaces]]*, in:

* [[Felix Hausdorff]], *[[Grundzüge der Mengenlehre]]*, Leipzig: Veit (1914), Reprinted by Chelsea Publishing Company (1944, 1949, 1965) \[ISBN:978-0-8284-0061-9, [ark:/13960/t2891gn8g](https://archive.org/details/grundzgedermen00hausuoft/page/n5/mode/2up)\]

The more general definition -- dropping Hausdorff's $T_2$-[[separation axiom]] and formulated in terms of [[closure operators]] that preserve finite [[unions]] -- is due to:

* [[Kazimierz Kuratowski]], *Sur l'opération Ā de l'Analysis Situs*, Fundamenta Mathematicae **3** (1922) 182–199 \[doi:10.4064/fm-3-1-182-199\]

The modern formulation via [[open set]] was widely popularized by:

* {#Bourbak71} [[Nicolas Bourbaki]]], *Eléments de mathématique II. Première partie. Les structures fondamentales de l’analyse. Livre III. Topologie générale. Chapitre I. Structures topologiques.* Actualités scientifiques et industrielles, vol. 858. Hermann, Paris (1940)

  *General topology*, Elements of Mathematics III, Springer (1971, 1990, 1995) \[doi:10.1007/978-3-642-61701-0\]

### Further

Further textbook accounts:

* {#Kelley55} [[John Kelley]], *General topology*, D. van Nostrand, New York (1955), reprinted as: Graduate Texts in Mathematics, Springer (1975) \[ISBN:978-0-387-90125-1\]

* {#Dugundji66} [[James Dugundji]], *Topology*, Allyn and Bacon 1966 ([pdf](https://www.southalabama.edu/mathstat/personal_pages/carter/Dugundji.pdf))

* {#Munkres75} [[James Munkres]], _Topology_, Prentice Hall (1975, 2000) \[ISBN:0-13-181629-2, [pdf](http://mathcenter.spb.ru/nikaan/2019/topology/4.pdf)\]

* [[Richard E. Hodel]] (ed.), *Set-Theoretic Topology*, Academic Press (1977) \[doi:10.1016/C2013-0-11355-4\]

* [[Klaus Jänich]], *Topology*, Undergraduate Texts in Mathematics, Springer (1984, 1999) \[ISBN:9780387908922, [doi:10.1007/978-3-662-10574-0](https://doi.org/10.1007/978-3-662-10574-0), Chapters 1-2: [pdf](http://topologicalmedialab.net/xinwei/classes/readings/Janich/Janich_Topology_ch1-2.pdf)\]

* {#Engelking89} [[Ryszard Engelking]], *General Topology*, Sigma series in pure mathematics **6**, Heldermann 1989 ([ ISBN 388538-006-4](https://www.heldermann.de/SSPM/SSPM06/sspm06.htm))

* {#Vickers89} [[Steven Vickers]], _Topology via Logic_, Cambridge University Press (1989) ([toc pdf](http://www.gbv.de/dms/ilmenau/toc/21309293X.PDF))

* {#Bredon93} [[Glen Bredon]], _Topology and Geometry_, Graduate texts in mathematics **139**, Springer 1993 ([doi:10.1007/978-1-4757-6848-0](https://link.springer.com/book/10.1007/978-1-4757-6848-0),  [pdf](http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf))

* {#Lawson03} Terry Lawson, _Topology: A Geometric Approach_, Oxford University Press (2003) ([pdf](http://users.metu.edu.tr/serge/courses/422-2014/supplementary/TGeometric.pdf))

* Anatole Katok, Alexey Sossinsky, *Introduction to Modern Topology and Geometry* (2010) \[toc pdf, [[KatokSossinsky-Topology-Ch1.pdf:file]]\]

and leading over to [[homotopy theory]]:

* [[Ioan Mackenzie James]], *General Topology and Homotopy Theory*, Springer 1984
([doi:10.1007/978-1-4613-8283-6](https://doi.org/10.1007/978-1-4613-8283-6))

On [[counterexamples]] in topology:

* [[Lynn Arthur Steen]], [[J. Arthur Seebach]], _[[Counterexamples in Topology]]_, Springer 1971 ([doi:10.1007/978-1-4612-6290-9](https://link.springer.com/book/10.1007/978-1-4612-6290-9))

With emphasis on [[category theory|category theoretic]] aspects of [[general topology]], notably on [$T_n$-reflections](separation+axioms#Reflection):

* {#HerrlichStrecker71} [[Horst Herrlich]], [[George Strecker]], *Categorical topology -- Its origins as exemplified by the unfolding of the theory of topological reflections and coreflections before 1971* ([pdf](https://link.springer.com/content/pdf/10.1007%2F978-94-017-0468-7_15.pdf)), pages 255-341 in: C. E. Aull, R Lowen (eds.), *Handbook of the History  of General Topology. Vol. 1*, Kluwer 1997 ([doi:10.1007/978-94-017-0468-7](https://link.springer.com/book/10.1007/978-94-017-0468-7))

* {#BradleyBrysonTerilla20} [[Tai-Danae Bradley]], [[Tyler Bryson]], [[John Terilla]], _Topology -- A categorical approach_, MIT Press 2020 ([ISBN:9780262539357](https://mitpress.mit.edu/books/topology), [web version](https://topology.pubpub.org/))

See also:

* [[Alan Hatcher]], _[Algebraic Topology](https://www.math.cornell.edu/~hatcher/AT/ATpage.html)_

* [[Neil Strickland]], *A Bestiary of Topological Objects* \[pdf, [[Strickland-BestiaryTopological.pdf:file]]\]

and see further references at *[[algebraic topology]]*.

Lecture notes:

* [[Friedhelm Waldhausen]], _Topologie_ ([pdf](https://www.math.uni-bielefeld.de/~fw/ein.pdf))

* Alex Kuronya, _Introduction to topology_, 2010 ([pdf](https://www.uni-
frankfurt.de/64271720/TopNotes_Spring10.pdf))

* Anatole Katok, Alexey Sossinsky, _Introduction to modern topology and geometry_ ([pdf](http://www.personal.psu.edu/axk29/MASS-07/Background-forMASS.pdf))

* {#Schreiber17} [[Urs Schreiber]], _[[Introduction to Topology]]_, Bonn 2017

* {#Mueger18} [[Michael Müger]], _Topology for the working mathematician_, Nijmegen 2018 ([pdf](https://www.math.ru.nl/~mueger/topology.pdf))

Basic topology set up in [[intuitionistic mathematics]] is discussed in

* {#Waaldijk96} [[Franka Waaldijk]], _modern intuitionistic topology_, 1996 ([pdf](http://www.fwaaldijk.nl/modern%20intuitionistic%20topology.pdf))

See also:

* [[Topospaces]], a Wiki with basic material on topology.


topology and geometry

<div class="rightHandSide toc">
[[!include mathematicscontents]]
</div>

up: [[mathematics|Mathematics]]

Some of the subfields of topology and geometry touched in the nlab: [[general topology]], [[homotopy theory]], [[rational homotopy theory]], [[differential geometry]] (including [[symplectic geometry]], [[synthetic differential geometry]]), [[differential topology]], [[algebraic geometry]], [[noncommutative algebraic geometry]], [[noncommutative geometry]].

Nowdays however there exist many frameworks in which in a unified way, the main objects (kinds of spaces, like schemes or supermanifolds) in several of the above categories can be obtained, for example [[ringed space]], [[ringed site]], [[structured (infinity,1)-topos]]...

Here are some nonexhaustive but longer grouped-lists of entries: [[geometry]], [[topology]].


topology of mapping spaces

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

#Contents#
* table of contents
{:toc}

## Idea ##

As shown in [[manifolds of mapping spaces]], the space of [[smooth map]]s from a [[sequentially compact space|sequentially compact]] [[Froelicher space|Frölicher]] (or [[diffeological space|diffeological]] or [[Chen space|Chen]]) space is again a smooth manifold.  As such, its [[topology]] can be determined by considering how the [[coordinate chart|chart]]s are glued together.

In some cases, there is an easy method available to study the topology.  If the target manifold, say $M$, [[embedding|embeds]] as a [[submanifold]] of some [[convenient vector space]], say $V$, then $C^\infty(S,M)$ embeds as a submanifold of $C^\infty(S,V)$.  If the original embedding is as a closed submanifold, so is the embedding of mapping spaces.  This means that it is possible to propagate topological results down from $C^\infty(S,V)$ to $C^\infty(S,M)$.  Furthermore, if $M$ is a [[deformation retract]] of a neighbourhood of its image in $V$, then so is $C^\infty(S,M)$ in $C^\infty(S,V)$.

## Embeddings ##

+--{: .num_theorem #embedding}
###### Theorem
Let $g \colon M \to N$ be an embedding of smooth manifolds.  Let $S$ be a sequentially compact Frölicher space.  Then $C^\infty(S,g) \colon C^\infty(S,M) \to C^\infty(S,N)$ is an embedding of smooth manifolds.
=--

## Related entries

* [[compact-open topology]]

* [[differential topology of mapping spaces]]

  * [[C-k topology]]


topology of uniform convergence

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[topological space|topology]] on [[mapping spaces]] from some [[set]] into a [[topological vector space]].

## Related concepts

* [[uniform convergence]]

* [[compact-open topology]]

## References

* Wikipedia, _[Topology of uniform convergence](https://en.wikipedia.org/wiki/Topology_of_uniform_convergence)_

[[!redirects topologies of uniform convergence]]


Topos

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
#### Category Theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

By __$Topos$__ (or __$Toposes$__) is denoted the [[category]] of [[topos]]es. Usually this means:

* [[object]]s are [[topos]]es;

* [[morphism]]s are [[geometric morphism]]s of toposes.

This is naturally a [[2-category]], where

* [[2-morphism]] are [[geometric transformation]]s

That is, a 2-morphism $f\to g$ is a [[natural transformation]] $f^* \to g^*$ (which is, by [[mate]] calculus, equivalent to a natural transformation $g_* \to f_*$ between [[direct image]]s).  Thus, $Toposes$ is equivalent to both of

* the (non-full) [[sub-2-category]] of $Cat^{op}$ on categories that are toposes and morphisms that are the inverse image parts of geometric morphisms, and
* the (non-full) sub-2-category of $Cat^{co}$ on categories that are toposes and morphisms that are the direct image parts of geometric morphisms.

### Related 2-categories

* There is also the sub-2-category $ShToposes = GrToposes$ of [[sheaf topos]]es (i.e. Grothendieck toposes).

* Note that in some literature this 2-category is denoted merely $Top$, but that is also commonly used to denote [[Top|the category]] of [[topological spaces]].

* We obtain a very different 2-category of toposes if we take the morphisms to be [[logical functors]]; this 2-category is sometimes denoted $Log$ or $LogTopos$.

## Properties

### From topological spaces to toposes

The operation of forming [[categories of sheaves]]

$$
  Sh(-) : Top \to ShToposes
$$

embeds [[topological space]]s into toposes. For $f : X \to Y$ a [[continuous map]] we have that $Sh(f)$ is the [[geometric morphism]]

$$
  Sh(f) : Sh(X) \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}
  Sh(Y)
$$

with $f_*$ the [[direct image]] and $f^*$ the [[inverse image]].

Strictly speaking, this functor is not an [[embedding]] if we consider $Top$ as a 1-category and $Toposes$ as a 2-category, since it is then not [[fully faithful]] in the 2-categorical sense---there can be nontrivial 2-cells between geometric morphisms between toposes of sheaves on topological spaces.

However, if we regard $Top$ as a [[(1,2)-category]] where the 2-cells are inequalities in the [[specialization ordering]], then this functor does become a 2-categorically full embedding (i.e. an equivalence on hom-categories) if we restrict to the full subcategory $SobTop$ of [[sober spaces]].  This embedding can also be extended from $SobTop$ to the entire category of [[locales]] (which can be viewed as "Grothendieck 0-toposes").

### From toposes to higher toposes

There are similar full embeddings $ShTopos \hookrightarrow Sh 2 Topos$ and $ShTopos \hookrightarrow Sh(n,1)Topos$ of sheaf (1-)toposes into [[2-sheaf]] [[2-topos]]es and sheaf [[(n,1)-topos]]es for $2\le n\le \infty$.  Note that these embeddings are *not* the identity functor on underlying categories: a 1-topos is not itself an $n$-topos, instead we have to take $n$-sheaves on a suitable generating [[site]] for it.

### From locally presentable categories to toposes
 {#AdjunctionToLocallyPresentable}

There is a canonical [[forgetful functor]] $ U : Topos \to$ [[Cat]] that lands, by definition, in the sub-2-category of [[locally presentable categories]] and [[functors]] which preserve all limits / are [[right adjoints]].

This [[2-functor]] has a [[2-adjunction|right 2-adjoint]] ([Bunge-Carboni](#BungeCarboni)).

### Limits and colimits
 {#Limits}

The 2-category $Topos$ is not all that well-endowed with [[limits]], but its [[slice categories]] are finitely complete [[2-limit|as 2-categories]], and $ShTopos$ is closed under finite limits in $Topos/Set$.  In particular, the [[terminal object]] in $ShToposes$ is the topos [[Set]] $\simeq Sh(*)$.

#### Colimits
 {#Colimits}

The supply with colimits is better:

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

All small (indexed) [[2-colimit]]s in $ShTopos$ exists and are computed as (indexed) [[2-limit]]s in [[Cat]] of the underlying [[inverse image]] functors.

=--

This appears as ([Moerdijk, theorem 2.5](#Moerdijk))

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

Let

$$
  \array{
    \mathcal{F} &\stackrel{p_2}{\to}& \mathcal{E}_2
    \\
    {}^{\mathllap{p_1}}\downarrow &\swArrow& \downarrow^{\mathrlap{f_2}}
    \\
    \mathcal{E}_1 &\underset{f_1}{\to}& \mathcal{E}
  }
$$

be a [[2-pullback]] in $Topos$ such that

* $f_1, f_2$ are both [[pseudomonic morphism]]s

* $\mathcal{E}_1 \coprod \mathcal{E}_2 \to \mathcal{E}$ is an [[effective epimorphism]];

then the diagram of [[inverse image]] functors

$$
  \array{
    \mathcal{F} &\stackrel{p_2^*}{\leftarrow}& \mathcal{E}_2
    \\
    {}^{\mathllap{p_1^*}}\uparrow &\swArrow& \uparrow^{\mathrlap{f_2^*}}
    \\
    \mathcal{E}_1 &\underset{f_1^*}{\leftarrow}& \mathcal{E}
  }
$$

is a 2-pullback in [[Cat]] and so by the [above](#ColimitsByInverseImageLimits) the original square is also a 2-pushout.

=--

This appears as theorem 5.1 in ([BungeLack](#BungeLack))

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

The 2-category $Topos$ is an [[extensive category]]. Same for toposes bounded over a base.

=--

This is in ([BungeLack, proposition 4.3](#BungeLack)).

#### Pullbacks
 {#Pullbacks}

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

Let

$$
  \array{
    && \mathcal{X}
    \\
    && \downarrow^{\mathrlap{(g^* \dashv g_*)}}
    \\
    \mathcal{Y}
    &\stackrel{(f^* \dashv f_*)}{\to}&
    \mathcal{Z}
  }
$$

be a [[diagram]] of [[toposes]]. Then its [[pullback]] in the [[(2,1)-category]] version of $Topos$ is computed, roughly, by the [[pushout]] of their [[sites]] of definition.

More in detail: there exist [[sites]] $\tilde \mathcal{D}$, $\mathcal{D}$, and $\mathcal{C}$ with [[finite limit]]s and [[morphisms of sites]]

$$
  \array{
    && \mathcal{D}
    \\
    && \uparrow^{\mathrlap{g}}
    \\
    \tilde \mathcal{D}
    &\stackrel{f}{\leftarrow}&
    \mathcal{C}
  }
$$

such that

$$
  \left(
  \array{
    && \mathcal{X}
    \\
    && \downarrow^{\mathrlap{(g^* \dashv g_*)}}
    \\
    \mathcal{Y}
    &\stackrel{(f^* \dashv f_*)}{\to}&
    \mathcal{Z}
  }
  \right)
  \,\,\,
   \simeq
  \,\,\,
  \left(
  \array{
    && Sh(\mathcal{D})
    \\
    && \downarrow^{\mathrlap{(Lan_g \dashv (-)\circ g)}}
    \\
    Sh(\tilde \mathcal{D})
    &\stackrel{(Lan_f \dashv (-)\circ f)}{\to}&
    Sh(\mathcal{C})
  }
  \right)
  \,.
$$

Let then

$$
  \array{
    \tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D} &\stackrel{f'}{\leftarrow}& \mathcal{D}
    \\
    {}^{\mathllap{g'}}\uparrow &\swArrow_{\simeq}& \uparrow^{\mathrlap{g}}
    \\
    \tilde \mathcal{D}
    &\stackrel{f}{\leftarrow}&
    \mathcal{C}
  }
  \,\,\,\,\,
  \in
  Cat^{lex}
$$

be the [[pushout]] of the underlying [[categories]] in the [[full subcategory]] [[Cat]]${}^{lex} \subset Cat$ of categories with finite limits.

Let moreover

$$
  Sh(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D})
  \hookrightarrow
  PSh(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D})
$$

be the [[reflective subcategory]] obtained by [[localization]] at the class of morphisms generated by the inverse image $Lan_{f'}(-)$ of the [[covering]]s of $\mathcal{D}$ and the inverse image $Lan_{g'}(-)$ of the coverings of $\tilde \mathcal{D}$.

Then

$$
  \array{
    Sh(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D})
    &\to& \mathcal{X}
    \\
    \downarrow && \downarrow^{\mathrlap{(g^* \dashv g_*)}}
    \\
    \mathcal{Y}
    &\stackrel{(f^* \dashv f_*)}{\to}&
    \mathcal{Z}
  }
$$

is a [[pullback]] square.

=--

This appears for instance as ([Lurie, prop. 6.3.4.6](#Lurie)).

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

For [[localic toposes]] this reduces to the statement of [[localic reflection]]: the pullback of toposes is given by the of the underlying [[locales]] which in turn is the [[pushout]] of the corresponding [[frames]].

=--

### Free loop spaces

The [[free loop space object]] of a topos in [[Topos]] is called the [[isotropy group of a topos]].

## Related concepts

* **Topos**

* [[(∞,1)Topos]]

## References

The characterization of colimits in $Topos$ is in

* [[Ieke Moerdijk]], _The classifying topos of a continuous groupoid. I_ Transaction of the American mathematical society Volume 310, Number 2, (1988) ([pdf](http://www.ams.org/journals/tran/1988-310-02/S0002-9947-1988-0973173-9/S0002-9947-1988-0973173-9.pdf))
{#Moerdijk}

The fact that $Topos$ is extensive is in

* [[Marta Bunge]], [[Steve Lack]], _van Kampen theorem for toposes_ ([ps](http://www.maths.usyd.edu.au/u/stevel/papers/vkt.ps.gz))
{#BungeLack}

Limits and colimits of toposes are discussed in 6.3.2-6.3.4 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_ .
 {#Lurie}

There this is discussed for for [[(∞,1)-topos]]es, but the statements are verbatim true also for ordinary toposes (in the [[(2,1)-category]] version of $Topos$).

The adjunction between toposes and locally presentable categories is discussed in

* [[Marta Bunge]], [[Aurelio Carboni]], _The symmetric topos_, Journal of Pure and Applied Algebra 105:233-249, (1995)
 {#BungeCarboni}

category: category

[[!redirects Topos]]
[[!redirects Topoi]]
[[!redirects Toposes]]