separation axioms

> This is about the [[separation axioms]] in [[topology]]. For the axiom in [[set theory]] also called "separation", see [[axiom of separation]].

---

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### Context
#### Topology
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[[!include topology - contents]]
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# The separation axioms
* table of contents
{: toc}

## Idea
 {#Idea}

The plain definition of _[[topological space]]_ happens to allow examples where distinct points or distinct subsets of the underlying set of a topological space appear as as more-or-less unseparable as seen by the topology on that set. In many applications one wants to exclude at least some of such degenerate examples from the discussion and instead focus on sufficiently [[nice topological spaces]]. The relevant conditions to be imposed on top of the plain [[axioms]] of a [[topological space]] are hence known as _separation axioms_.

These axioms are all of the form of saying that two [[subsets]] (of certain forms) in the topological space are 'separated' from each other in one sense if they are 'separated' in a (generally) weaker sense. Most of these conditions can be expressed as [[separation axioms in terms of lifting properties|lifting properties with respect to maps of finite topological spaces or the real line]]. For example the weakest axiom (called $T_0$) demands that if two points are distinct as elements of the [[underlying]] set, then there exists at least one [[open subset]] that contains one but not the other,
i.e. the function determined by two distinct points is not [[continuous function|continuous]] as a map from the [[indiscrete space]]. In other words, $T_0$ says that each continuous function from the indiscrete space is necessarily trivial.
As a [[lifting property]] this is expressed as: the map collapsing the indiscrete space with two points into a single point, has the left lifting property with respect to the map collapsing the underlying space into a single point.

In this fashion one may impose a hierarchy of stronger axioms. For example demanding that given two distinct points, then each of them is contained in some open subset not containing the other ($T_1$), i.e.
any map from the space with one open point and one closed is necessarily trivial, or that such a pair of open subsets around two distinct points may in addition be chosen to be disjoint ($T_2$); such a pair of open subsets can be equivalently described as a continuous function to the space with two open points and one closed, and $T_2$ as saying that the inclusion of two open points into the space with two points and one closed, factors via any injective function from the space with two points, to the underlying space.
This last condition, $T_2$, also called the _[[Hausdorff topological space|Hausdorff condition]]_ is the most common among all separation axioms. Often (but by far not always) this is considered by default.

As a lifting property, $T_1$ is expressed as: the map collapsing the space with one point open and one point closed into a single point, has the left lifting property with respect to the map collapsing the underlying space into a single point. $T_2$ is expressed as: any injective map from the discerete space with two points into the space, has the left lifting proprety with respect to the map collapsing the space with one closed point and
two open points into a single point.

Rewriting the [[separation axioms in terms of lifting properties]]
with respect to maps of finite topological spaces, i.e. monotone maps of preorders, provides a combinatorial point of view on this hierachy,
and simplifies certain universal constructions such as
[reflection](#Reflection) or  [Kolmogorov quotient](#KolmogorovQuotient).

The main separation axioms are these:

{#TableOfMainSeparationAxioms}
[[!include main separation axioms -- table]]
{#TableOfMainSeparationAxiomsAsLiftingProperties}
[[!include main separation axioms -- as lifting properties]]

Originally in [Tietze 23](#Tietze23) the four separation axioms $T_2, T_3, T_4, T_5$ were considered (see at _[History](#History)_ below for more); nowadays one considers various more. Besides the extrapolation of the original sequence from $T_0$ through $T_6$ (with $T_{2\frac{1}{2}}$ and $T_{3\frac{1}{2}}$ interpolated), there is a similar sequence of axioms called $R_0, R_1, R_2, R_3$ (with their extrapolations and interpolations) of the same form, except that they do not start with mentioning two set-theoretically distinct points, but two points satisfying the conclusion of $T_0$. This and more is spelled out [below](#TheClassicalTheory).

There are also axioms that do not follow the pattern of "if certain two subsets are separated in some weak sense, then they are also separated in some stronger sense", but that still axiomatize some kind of separatedness. For example the condition on a [[topological space]] being [[sober topological space|sober]] is of a different nature, but is implied by $T_2$ and implies $T_0$. Notice that via their [[full subcategory|full embedding]] into [[locales]], [[sober topological spaces]] may be understood without reference to their underlying set of points at all.

All separation axioms are satisfied by [[metric spaces]], from whom the concept of topological space was originally abstracted. Hence imposing some of them may also be understood as gauging just how far one allows topological spaces to generalize away from metric spaces.

Several separation axioms may also be interpreted in broader contexts that plain topological spaces, for instance for [[convergence space]] or for [[locales]]; or the may be considered under weaker assumptions, such as those of [[constructive mathematics]] and [[predicative mathematics]].

## The classical theory
 {#TheClassicalTheory}

First, we will consider how, for topological spaces in [[classical mathematics]], the separation axioms are about sets\' being 'separated' as stated above.  Throughout, fix a [[topological space]] $S$.

### Separation conditions

Fix two sets ([[subsets]]) $F$ and $G$ of $S$.

*  The sets $F$ and $G$ are __[[disjoint sets|disjoint]]__ if their [[intersection]] is [[empty set|empty]]:
   $$ F \cap G = \empty .$$
*  They are __topologically disjoint__ if there exists a [[neighbourhood]] of one set that is disjoint from the other set:
   $$ (\exists\; U \stackrel{\circ}\supseteq F,\; U \cap G = \empty) \;\vee\; (\exists\; V \stackrel{\circ}\supseteq G,\; F \cap V = \empty) .$$
   Notice that topologically disjoint sets must be disjoint.
*  They are __separated__ if each set has a neighbourhood that is disjoint from the other set:
   $$ (\exists\; U \stackrel{\circ}\supseteq F,\; U \cap G = \empty) \;\wedge\; (\exists\; V \stackrel{\circ}\supseteq G,\; F \cap V = \empty)
   \;\;\equiv\;\;
   \exists\; U \stackrel{\circ}\supseteq F,\; \exists\; V \stackrel{\circ}\supseteq G,\; U \cap G = \empty \;\wedge\; F \cap V = \empty .$$
   Notice that separated sets must be topologically disjoint.

*  {#SeparatedByNeighbourhoods} They are __separated by neighbourhoods__ if they have disjoint neighbourhoods:
   $$ \exists\; U \stackrel{\circ}\supseteq F,\; \exists\; V \stackrel{\circ}\supseteq G,\; U \cap V = \empty .$$
   Notice that sets separated by neighbourhoods must be separated.

*  They are __separated by closed neighbourhoods__ if they have disjoint closed neighbourhoods:
   $$ \exists\; U \stackrel{\circ}\supseteq F,\; \exists\; V \stackrel{\circ}\supseteq G,\; Cl(U) \cap Cl(V) = \empty .$$
   Notice that sets separated by closed neighbourhoods must be separated by neighbourhoods.
*  They are __separated by a function__ if there exists a continuous [[real number|real]]-valued [[function]] on the space that maps $F$ to $0$ and $G$ to $1$:
   $$ \exists\; f: S \to \mathbf{R},\; F \subseteq f^*(\{0\}) \;\wedge\; G \subseteq f^*(\{1\}) .$$
   Notice that sets separated by a function must be separated by closed neighbourhoods (the preimages of $[-\epsilon, \epsilon]$ and $[1-\epsilon, 1+\epsilon]$).
*  Finally, they are __precisely separated by a function__ if there exists a continuous real-valued function on the space that maps precisely $F$ to $0$ and $G$ to $1$:
   $$ \exists\; f: S \to \mathbf{R},\; F = f^*(\{0\}) \;\wedge\; G = f^*(\{1\}) .$$
   Notice that sets precisely separated by a function must be separated by a function.

Often $F$ and $G$ will be points (identified with their [[singleton]] subsets); in that case, one usually says _distinct_ in place of _disjoint_.

Often $F$ or $G$ will be closed sets; notice that disjoint closed sets are automatically separated, while a closed set and a point, if disjoint, are automatically topologically disjoint.

### Separation axioms

The classical separation axioms are all statements of the form

*  When $F$ is a (point/closed) set and $G$ is a (point/closed) set, if $F$ and $G$ are (separated in some weak sense), then they are (separated in some strong sense).

The axioms with names (at least with known to the authors so far of this article) are summarised in the tables below.  When a row or column is missing from a table, either no name is known or the implication follows from the converses mentioned after the separation conditions above in the context of that table; there are two potential tables that are completely blank for the latter reason.  When an entry in a table is repeated, that corresponds to a theorem that one separation axiom implies another.

When both sets are points:

Stronger condition ↓\Weaker condition →	 Distinct	 Topologically distinct

Topologically distinct	 $T_0$

Separated	 $T_1$	 $R_0$

Separated by neighbourhoods	 $T_2$	 $R_1$

Separated by closed neighbourhoods	 $T_{2\frac{1}{2}}$	 $R_{1\frac{1}{2}}$

Separated by a function	 Completely $T_2$	 Completely $R_1$

When one set is a point and the other is closed:

Stronger condition ↓\Weaker condition →	 Disjoint

Separated by neighbourhoods	 Regular

Separated by closed neighbourhoods	 Regular

Separated by a function	 Completely regular

When both sets are closed:

Stronger condition ↓\Weaker condition →	 Disjoint

Separated by neighbourhoods	 Normal

Separated by closed neighbourhoods	 Normal

Separated by a function	 Normal

Precisely separated by a function	 Perfectly normal

When the sets are arbitrary:

Stronger condition ↓\Weaker condition →	 Separated

Separated by neighbourhoods	 Completely normal ($T_5$)

### Reformulation in terms of topological closures
 {#EquivalentIncarnationsOfTheAxioms}

Many of the separation axioms have a useful equivalent formulation in terms of certain  [[topological closures]].

+-- {: .num_prop #T0InTermsOfClosureOfPoints}
###### Proposition
**($T_0$ in terms of topological closures)**

A [[topological space]] $(X,\tau)$ is $T_0$
precisely if the function $Cl(\{-\})$
from the underlying set of $X$ to the set of [[irreducible closed subsets]] of $X$, is [[injective function|injective]]:

$$
  Cl(\{-\}) \;\colon\; X \hookrightarrow IrrClSub(X)
$$

=--

(This statement also motivates the definition of [[sober topological spaces]], for which $Cl(\{-\})$ is required to be a [[bijection]]).

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

Assume first that $X$ is $T_0$. Then we need to show that if $x,y \in X$ are such that
$Cl(\{x\}) = Cl(\{y\})$ then $x = y$. Hence assume that $Cl(\{x\}) = Cl(\{y\})$.
Since the closure of a point is the [[complement]] of the union of the open subsets not containing the point
(lemma \ref{UnionOfOpensGivesClosure}),
this means that the union of open subsets that do not contain $x$
is the same as the union of open subsets that do not contain $y$:

$$
  \underset{ {U \subset X \, \text{open}} \atop { U \subset X\backslash \{x\} } }{\cup} \left( U \right)
  \;=\;
  \underset{ {U \subset X \, \text{open}} \atop { U \subset X\backslash \{y\} } }{\cup} \left( U \right)
$$

But if the two points were distinct, $x \neq y$, then by $T_0$ one of the above unions would contain $x$ or $y$, while the other would not, in contradiction to the above equality. Hence we have a [[proof by contradiction]].

Conversely, assume that $\left( Cl\{x\} = Cl\{y\}\right) \Rightarrow \left( x = y\right)$, and assume that $x \neq y$.
Hence by [[contraposition]] $\mathrm{Cl}(\{x\}) \neq \mathrm{Cl}(\{y\})$. We need to show that there exists
an open set which contains one of the two points, but not the other.

Assume there were no such open subset. By lemma [this lemma](Introduction+to+Topology+--+1#UnionOfOpensGivesClosure) this would mean
that $x \in \mathrm{Cl}(\{y\})$ and that $y \in \mathrm{Cl}(\{x\})$. But this would imply that
$Cl(\{x\}) \subset \mathrm{Cl}(\{y\})$ and that $\mathrm{Cl}(\{y\}) \subset \mathrm{Cl}(\{x\})$,
hence that $\mathrm{Cl}(\{x\}) = \mathrm{Cl}(\{y\})$. This is a [[proof by contradiction]].

=--

+-- {: .num_prop #T1InTermsOfClosureOfPoints}
###### Proposition
**($T_1$ in terms of topological closures)**

A [[topological space]] $(X,\tau)$ is $T_1$ precisely if
all its points are [[closed points]].

=--

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

Assume first that $(X,\tau)$ is $T_1$. We need to show that for every point $x \in X$ we have
$Cl(\{x\}) = \{x\}$. Since the closure of a point is the [[complement]] of the union of all open subsets not containing this point,
this is the case precisely if the union of all open subsets not containing $x$ is $X \backslash \{x\}$, hence
if every point $y \neq x$ is member of at least one open subset not containing $x$. This is true by $T_1$.

Conversely, assume that for all $x \in X$ then $Cl(\{x\}) = \{x\}$. Then for $x \neq y \in X$ two
distinct points we need to produce an open subset of $y$ that does not contain $x$. But as before,
since $Cl(\{x\})$ is the complement of the union of all open subsets that do not contain $x$, and
the assumption $Cl\{x\} = \{x\}$ means that $y$ is member of one of these open subsets that do not
contain $x$.

=--

+-- {: .num_prop #T2InTermsOfClosedDiagonal}
###### Proposition
**($T_2$ in terms of topological closures)**

A [[topological space]] $(X,\tau_X)$ is $T_2$=[[Hausdorff topological space|Hausdorff]]
precisely if the [[image]] of the [[diagonal]]

$$
  \array{
     X &\overset{\Delta_X}{\longrightarrow}& X \times X
     \\
     x &\overset{\phantom{AAA}}{\mapsto}& (x,x)
  }
$$

is a [[closed subset]] in the [[product topological space]]
$(X \times X, \tau_{X \times X})$.

=--

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

The Hausdorff condition, that for $x \neq y \in X$ then there exist disjoint open neighbourhood $U_x, U_y \subset X$, is equivalently rephrased in terms of the product topology as: Every point $(x,y) \in X$ which is not on the diagonal has an open neighbourhood $U_x \times U_y$ which still does not intersect the diagonal.

Hence if $X$ is Hausdorff, then the diagonal $\Delta_X(X) \subset X \times X$ is the complement of a union of such open sets, and hence is closed.

Conversely, if the diagonal is closed, then (by [this lemma](Introduction+to+Topology+--+1#UnionOfOpensGivesClosure)) every point $(x,y)$ not on the diagonal, hence with $x \neq y$, has an open neighbourhood $U_x \times U_y$ still not intersecting the diagonal, hence so that $U_x \cap U_y = \emptyset$. Thus $(X,\tau)$ is Hausdorff.

=--

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

The characterization of the Hausdorff separation condition via the closure of the diagonal in prop. \ref{T2InTermsOfClosedDiagonal} is the basis for the definition of _[[separated scheme]]_.

=--

+-- {: .num_prop #T3InTermsOfTopologicalClosures}
###### Proposition
**($T_3$ in terms of topological closures)**

A [[topological space]] $(X,\tau)$ is [[regular topological space|regular]], precisely if for all [[closed subsets]] $x \in X$ with [[open neighbourhood]] $U \supset \{x\}$ there exists a smaller open neighbourhood $V \supset \{x\}$ whose [[topological closure]] $Cl(V)$ is still contained in $U$:

$$
  \{x\} \subset V \subset Cl(V) \subset U
  \,.
$$

=--

The **proof** of prop. \re{T3InTermsOfTopologicalClosures} is the direct  specialization of the following proof for prop. \ref{T4InTermsOfTopologicalClosures} to the case that $C = \{x\}$ (using that by $T_1$, which is part of the definition of $T_3$, the singleton subset is indeed closed by prop. \ref{T1InTermsOfClosureOfPoints}).

+-- {: .num_prop #T4InTermsOfTopologicalClosures}
###### Proposition
**($T_4$ in terms of topological closures)**

A [[topological space]] $(X,\tau)$ is [[normal topological space|normal]], precisely if for all [[closed subsets]] $C \subset X$ with [[open neighbourhood]] $U \supset C$ there exists a smaller open neighbourhood $V \supset C$ whose [[topological closure]] $Cl(V)$ is still contained in $U$:

$$
  C \subset V \subset Cl(V) \subset U
  \,.
$$

=--

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

In one direction, assume that $(X,\tau)$ is normal, and consider $C \subset U$. It follows that the [[complement]] of the open subset $U$ is closed and disjoint from $C$:

$$
  C \cap X \backslash U = \emptyset
  \,.
$$

Therefore by assumption of normality of $(X,\tau)$, there exists open neighbourhoods $V \supset C$ and $W \supset X \backslash U$ with

$$
  V \cap W = \emptyset
  \,.
$$

But this means that

$$
  V \subset X \backslash W
$$

and since the [[complement]] $X \backslash W$ of the open set $W$ is closed, it still contains the closure of $V$, so that we have

$$
  C \subset V \subset Cl(V) \subset X \backslash W \subset U
  \,.
$$

In the other direction, assume that for every open neighbourhood $U \supset C$ of a closed subset $C$ there exists a smaller open neighbourhood $V$ with $C \subset V \subset Cl(V) \subset U$. Consider disjoint closed subsets $C_1, C_2 \subset X$. We need to produce disjoint open neighbourhoods for them.

From their disjointness it follows that $X \backslash C_2 \supset C_1$ is an open neighbourhood. Hence by assumption there is an open neighbourhood $V$ with

$$
  C_1 \subset V \subset Cl(V) \subset X \backslash C_2
  \,.
$$

Thus $V \supset C_1$ and $X \backslash Cl(X) \supset C_2$ are two disjoint open neighbourhoods, as required.

=--

### Relations between the axioms

First of all, notice (prop. \ref{T1InTermsOfClosureOfPoints}) that the $T_1$ condition, saying that distinct points are separated, is equivalent to the condition that every point is closed.  Thus, $T_1$ serves as a linchpin between conditions on points and conditions on closed sets.

Many implications between separation axioms can be seen in the following Hasse diagram:

Here, there are two entries at each node; the one on the right includes the $T_0$ axiom, while the one on the left does not.  This diagram shows the separation axioms as a meet sub-[[semilattice]] of the lattice of all conditions on topological spaces; for example, you can see, by following the diagram upwards, that any space that is both **n**ormal and **r**egular must be $R_3$.  And since $R_3$ never appears in the tables above, you can take this as a *definition* of $R_3$.

In general, the names in this diagram are:

*  'P' for 'perfectly',
*  'C' for 'complete',
*  'N' for '[[normal space|normal]]',
*  'R' (without a subscript) for '[[regular space|regular]]'
*  'T' or 'R' with a subscript are written and pronounced that way.

Warning: $T_i$ for $i \geq 3$ has been used in different ways in the past, and perhaps by some schools still.  Also, all of the $R_i$ terms are rare.  It is safest to say, for example, 'normal Hausdorff' for $T_4$ and clearer to say, for example, 'normal regular' for $R_3$.  If you want to avoid the subscript terms entirely, then you can, by doing the above and the following:

* $T_{2\tfrac{1}{2}}$ = [[Urysohn space|Urysohn]]
*  $T_2$ = [[Hausdorff space|Hausdorff]],
*  $T_1$ = accessible,
*  $T_0$ = [[Kolmogorov space|Kolmogorov]],
*  $R_1$ = reciprocal,
*  $R_0$ = [[symmetric topological space|symmetric]].

On the other hand, if you want to use *more* symbols, then you can:

*  $R_2$ = regular,
*  $R_{2\frac{1}{2}}$ = completely regular.

It would be easy to invent an $N_i$ series for the various kinds of normal spaces, but nobody seems to have done so yet.

Other terms are also in use, principally 'Tychonoff' for [[Tychonoff space|completely regular Hausdorff]] ($T_{3\frac{1}{2}}$).

### Reflection
 {#Reflection}

(cf. [Herrlich & Strecker 1971](#HerrlichStrecker71))

+-- {: .num_prop #HausdorffReflection}
###### Proposition
**($T_n$-reflection)**

Let $n \in \{0,1,2\}$. Then for every [[topological space]] $X$ there exists
a $T_n$-topological space $T_n X$ and a [[continuous function]] of the forma

$$
  t_n(X)
    \;\colon\;
  X \longrightarrow T_n X
$$

which is the  "closest approximation from the left" to $X$ by a $T_n$-topological space, in that
for $Y$ any $T_n$-space, then [[continuous functions]] of the form

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

are in [[bijection]] with [[continuous function]] of the form

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

and such that the bijection is constituted by

$$
  f = \tilde f \circ t_n(X)
  \;\colon\;
    X
      \overset{t_n(X)}{\longrightarrow}
    T_n X
      \overset{\tilde f}{\longrightarrow}
    Y
  \,.
$$

Here $X \overset{t_n(X)}{\longrightarrow} T_n(X)$ is called the _$T_n$-reflection_ of $X$.

* For $n = 0$ this is known as the _[[Kolmogorov quotient]]_ construction (see prop. \ref{KolmogorovQuotient} below).

* For $n = 2$ this is known as _[[Hausdorff reflection]]_ or _Hausdorffication_ or similar.

Moreover, the operation $T_n(-)$ extends to [[continuous functions]] $f \colon X \to Y$

$$
  (X \overset{f}{\to} Y)
    \;\mapsto\;
  (T_n X \overset{T_n f}{\to} T_n Y)
$$

such as to preserve [[composition]] of functions as well as [[identity functions]]:

$$
  (T_n g) \circ (T_n f) = T_n(g \circ f)
  \phantom{AA}
  \,,
  \phantom{AA}
  T_n (id_X) = id_{T_n X}
  \,
$$

Finally, the comparison map is compatible with this in that for all continuous functions $f \colon X \to Y$ then

$$
  t_n(Y) \circ f
   =
  T_n(f)\circ t_n(X)
$$

hence then follows [[commuting squares|squares commutes]]:

$$
  \array{
      X &\overset{f}{\longrightarrow}& Y
      \\
      {}^{\mathllap{t_n(X)}}\downarrow && \downarrow^{\mathrlap{t_n(Y)}}
      \\
      T_n X &\underset{T_n f}{\longrightarrow}& T_n Y
  }
  \,.
$$

=--

We give a **proof** of the existence of this reflection below as the proof of prop. \ref{HausdorffReflectionViaHomsIntoAllHausdorffSpaces}.

+-- {: .num_remark }
###### Remark
**([[reflective subcategories]])**

In the language of [[category theory]] the $T_n$-reflection of prop. \ref{HausdorffReflection}
says that

1. $T_n(-)$ is a _[[functor]]_ $T_n \;\colon\; Top \longrightarrow Top_{T_n}$ from the [[category]] [[Top]] of [[topological spaces]] to the [[full subcategory]] $Top_{T_n} \overset{\iota}{\hookrightarrow} Top$ of Hausdorff topological spaces;

1. $t_n(X) \colon X \to T_n X$ is a _[[natural transformation]]_ from the [[identity functor]] on [[Top]] to the functor $\iota \circ T_n$

1. $T_n$-topological spaces form a [[reflective subcategory]] of all [[topological spaces]] in that $T_n$ is [[left adjoint]] to the inclusion functor $\iota$; this situation is denoted as follows:

   $$
     Top_{T_n}
       \underoverset{\underset{\iota}{\hookrightarrow}}{\overset{H}{\longleftarrow}}{\bot}
     Top
     \,.
   $$

Generally, an _[[adjoint functor|adjunction]]_ between two [[functors]]

$$
  L \;\colon\; \mathcal{C} \leftrightarrow \mathcal{D} \;\colon\; R
$$

is for all pairs of objects $c \in \mathcal{C}$, $d \in \mathcal{D}$ a [[bijection]] between sets of [[morphisms]] of the form

$$
  \left\{
     L(c) \longrightarrow d
  \right\}
  \simeq
  \left\{
    c \longrightarrow R(d)
  \right\}
  \,.
$$

i.e.

$$
  Hom_{\mathcal{D}}(L(c), d)
  \underoverset{\simeq}{\phi_{c,d}}{\longrightarrow}
  Hom_{\mathcal{C}}(c, R(d))
$$

and such that these bijections are "[[natural bijection|natural]]" in that they for all pairs of morphisms $f \colon c' \to c$ and $g \colon d \to d'$ then the folowing [[commuting diagram|diagram commutes]]:

$$
  \array{
    Hom_{\mathcal{D}}(L(c), d)
      &\underoverset{\simeq}{\phi_{c,d}}{\longrightarrow}&
    Hom_{\mathcal{C}}(c, R(d))
    \\
    {\mathllap{g \circ (-) \circ L(f)}}\downarrow
       &&
    \downarrow{\mathrlap{ R(g) \circ (-) \circ f }}
    \\
    Hom_{\mathcal{C}}(L(c'), d')
     &\underoverset{\simeq}{\phi_{c',d'}}{\longrightarrow}&
    Hom_{\mathcal{D}}(c', R(d'))
  }
  \,.
$$

=--

There are various ways to see the existence and to construct the $T_n$-reflections. The following is the quickest way to see the existence, even if to some tastes the construction seems more implicit or abstract than the previous one.

+-- {: .num_prop #HausdorffReflectionViaHomsIntoAllHausdorffSpaces}
###### Proposition
**($T_n$-reflection via surjections into $T_n$-spaces)**

Let $n \in \{0,1,2\}$. Let $(X,\tau)$ be a [[topological space]] and consider the [[equivalence relation]] $\sim$ on the underlying set $X$
for which $x \sim y$ precisely if for every [[surjective function|surjective]] [[continuous function]] $f \colon X \to Y$ into any
$T_n$-topological space $Y$ we have $f(x) = f(y)$.

Then the set of [[equivalence classes]]

$$
  T_n X \coloneqq X /{\sim}
$$

equipped with the [[quotient topology]] is a $T_n$-topological space, and the quotient map $t_n(X) \;\colon\; X \to X/{\sim}$ exhibits the $T_n$-reflection of $X$, according to prop. \ref{HausdorffReflection}.

=--

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

First we observe that every continuous function $f \colon X \longrightarrow Y$ into a $T_n$-topological space $Y$
factors uniquely via $t_n(X)$ through a continuous function $\tilde f$

$$
  f = \tilde f \circ h_X
$$

where

$$
  \tilde f \colon [x] \mapsto f(x)
  \,.
$$

To see this, first factor $f$ through its [[image]] $f(X)$

$$
  f \;\colon\; X \longrightarrow f(X) \hookrightarrow Y
$$

equipped with its [[topological subspace|subspace topology]] as a subspace of $Y$. It follows that $f(X)$ is a $T_n$-topological space if $Y$ is.

It follows by definition of $t_n(X)$ that the factorization exists at the level of sets as stated,
since if $x_1, x_2 \in X$ have the same [[equivalence class]] $[x_1] = [x_2]$ in $T_n X$, then
by definition they have the same image under all continuous surjective functions to a $T_n$-space, hence in particular
under $X \to f(X)$. This means that $\tilde f$ as above is well defined. Moreover, it is clear that this is the unique factorization.

To see that $\tilde f$ is continuous, consider $U \in Y$ an open subset. We need to show that $\tilde f^{-1}(U)$ is open in $X/\sim$. But by definition of the [[topological quotient space|quotient topology]], this is open precisely if its pre-image under the quotient projection $t_n(X)$ is open, hence precisely if

$$
  (t_n(X))^{-1}(\tilde f^{-1}(U))
  =
  ( \tilde f \circ t_n(X) )^{-1}(U)
  =
  f(U)
$$

is open in $X$. But this is the case by the assumption that $f$ is continuous.

What remains to be seen is that $T_n X$ as constructed is indeed a $T_n$-topological space.
Hence assume that $[x] \neq [y] \in T_n X$ are two distinct points. We need to open neighbourhoods
around one or both of these point not containing the other point and possibly disjoint to each other.

Now by definition of $T_n X$ this means that there exists a $T_n$-topological space $Y$ and a surjective continuous function
$f \colon X \longrightarrow Y$ such that $f(x) \neq f(y) \in Y$. Accordingly, since $Y$ is $T_n$,
there exist the respective kinds of neighbourhoods around these image points in $Y$. Moreover, by the previous statement there
exists a continuous function $\tilde f \colon T_n X \to Y$ with $\tilde f([x]) = f(x)$ and $\tilde f([y]) = f(y)$.
By the nature of continuous functions,
the pre-images of these open neighbourhoods in $Y$ are still open in $X$ and still
satisfy the required disjunction properties. Therefore $T_n X$ is a $T_n$-space.

=--

Here are alternative constructions of the reflections:

+-- {: .num_prop #KolmogorovQuotient}
###### Proposition
**([[Kolmogorov quotient]])**

Let $(X,\tau)$ be a [[topological space]]. Consider the [[relation]] on the underlying set
by which $x_1 \sim x_1$ precisely if neighther $x_i$ has an [[open neighbourhood]] not containing the other.
This is an [[equivalence relation]]. The [[quotient topological space]] $X \to X/\sim$ by this
equivalence relation  exhibits the $T_0$-reflection of $X$ according to prop. \ref{HausdorffReflection}.

=--

+-- {: .num_prop #HausdorffReflectionViaTransitiveClosureOfDiagonal}
###### Proposition
**([[Hausdorff reflection]])**

For $(Y,\tau_Y)$ a [[topological space]], write $r_Y \subset Y \times Y$
for the [[transitive relation|transitive closure]] of tthe [[relation]] given by the [[topological closure]] $Cl(\Delta_Y)$ of the [[image]] of the [[diagonal]] $\Delta_Y \colon Y \hookrightarrow Y \times Y$.

$$
  r_Y \coloneqq Trans(Cl(Delta_Y))
  \,.
$$

Now for $(X,\tau_X)$ a [[topological space]], define by [[induction]] for each [[ordinal number]] $\alpha$ an [[equivalence relation]] $r^\alpha$ on $X$ as follows, where we write $q^\alpha \colon X \to H^\alpha(X)$ for the corresponding [[quotient topological space]] projection:

We start the induction with the trivial equivalence relation:

* $r^0_X \coloneqq \Delta_X$;

For a [[successor ordinal]] we set

* $r_X^{\alpha+1} \coloneqq \left\{ (a,b) \in X \times X  \,\vert\, (q^\alpha(a), q^\alpha(b)) \in r_{H^\alpha(X)} \right\}$

and for a [[limit ordinal]] $\alpha$ we set

* $r_X^\alpha \coloneqq \underset{\beta \lt \alpha}{\cup} r_X^\beta$.

Then:

1. there exists an ordinal $\alpha$ such that $r_X^\alpha = r_X^{\alpha+1}$

1. for this $\alpha$ then $H^\alpha(X) = H(X)$ is the Hausdorff reflection from prop. \ref{HausdorffReflectionViaHomsIntoAllHausdorffSpaces}.

=--

([vanMunster 14, section 4](Hausdorff+space#vanMunster14))

### Other axioms

There are other axioms sometimes included among the separation axioms that don\'t fit the preceding pattern; but like the others, they all hold of a metric space:

*  [[sober space|sober]] and having enough points,
*  [[semiregular space|semiregular]],
*  [[fully normal topological space|fully normal]] and fully $T_4$, which are related to [[paracompact space|paracompactness]] (see at _[[fully normal spaces are equivalently paracompact]]_).

## Beyond the classical theory

The axioms $T_1$ and below can be phrased entirely in terms of the [[specialisation order]], as follows:

*  In general, the specialisation order is a [[preorder]].
*  The space is $T_0$ if and only if the specialisation order is a [[partial order]].
*  The space is $R_0$ if and only if the specialisation order is an [[equivalence relation]].
*  The space is $T_1$ if and only if the specialisation order is the [[equality relation]].

Note that *any* preorder is the specialisation order for its own [[specialisation topology]].

The separation conditions that appear in $T_2$ and below, or rather their [[negation]]s, can be easily phrased in terms of the [[convergence space|convergence structure]], as follows:

*  Two points are not distinct if and only if they are equal (of course).
*  They are __topologically indistinguishable__ (that is, not topologically distinct) if and only if every [[net]] (or [[filter]]) that [[convergence|converges]] to one must also converge to the other; it\'s enough to check the [[ultrafilters]] generated by the two points.
*  They are not separated if and only there exists a net (or proper filter) that converges to both.

So by taking contrapositives, it\'s easy to generalise $T_2$ and below to [[convergence spaces]].  (All of the axioms *can* be generalised to convergence spaces, since the convergence structure determines the topology, but there are several ways to do so, and it\'s not clear in general which is best.)

For [[locales]], the axioms at the other end are clearest.  Here we want to put everything in terms of open sets, so we simply work with the [[complements]] of the closed sets that appear in those axioms.  Rather than talk about a closed set $F$ and a neighbourhood $U$ of $F$, we talk about an open set $G$ and an open set $U$ such that $G \cup U$ is the entire space.  Now the axioms at the low end are tricky, although there is a standard answer as far down as $T_2$.  (Note that every locale is $T_0$, indeed sober.)

In [[constructive mathematics]], while the classical definitions all make sense, they are never quite what is wanted.  For the low axioms, one may use, as with convergence spaces, conditions that are classically the negations of the separation conditions; for the high axioms, one may use the open sets that are classically the complements of the closed sets in the axioms.  In the middle axioms, these work together; for example, the condition that a point $x$ is disjoint from a closed set $F$ becomes the condition that $x$ belongs to an open set $G$.

Specific examples should be found on the pages for specific separation axioms.

## History
 {#History}

In [Tietze 23, part B, starting on page 300](#Tietze23). 4 axioms are discussed, called (in words, not numbers) the first, second, third, and fourth separation axioms (*erstes, zweites, drittes, und viertes Trennbarkeitsaxiom*).  The first of these is $T_2$, the second is $T_3$, the third is $T_4$, and the fourth is $T_5$.  So while this paper may be the first to consider a hierarchy of separation axioms, it is not the source of our $T_i$ notation, and it does not number them in the same way.

All of these after the first are stated in such a way as to not imply the first, and those after the second are similarly stated in such a way as not to imply the second.  However, Tietze does seem to want them to be a hierarchy.  For one thing, his general definition of topological space — stated in multiple equivalent ways in part A — includes the first separation axiom, so in context it seems that the others are meant to be postulated only of spaces that are already Hausdorff.

Also, after stating the axioms, he immediately provides examples of spaces that satisfy one property but not a higher one, taking care to list both the first and the second separation axiom among those that the third is independent of, but leaving out the second, listing only the first and the third, when listing the axioms that the fourth is independent of.  He does eventually give an example of a space that satisfies the second and third but not the first (so a normal regular space that is not Hausdorff), but it is later and more of an afterthought (and the only non-Hausdorff space in this paper).  He never asks whether there exists of a regular space that is not normal.

Immediately after this example of a non-Hausdorff space, [Tietze 23](#Tietze23) lists the $T_1$ axiom after all!  But it is not on the same level as the others to him.  Instead, it is merely an alternative form of the first separation axiom that may be used in the presence of the second.  It still appears that non-Hausdorff spaces are not considered to be real topological spaces worthy of one\'s attention.

## Related concepts

* [[nice topological space]]

* [[separation axioms in terms of lifting properties]]

## References

An original article is

* {#Tietze23} Heinrich Tietze, _Beitrage zur allgemeinen Topologie. I. Axiome für verschiedene Fassungen des Umgebungsbegriffs_, Mathematische Annalen, vol 88, pages 290-311 (1923) ([online scan](https://www.digizeitschriften.de/en/dms/img/?PID=GDZPPN00226921X&physid=phys298#navi))

Lecture notes:

* {#Naik} [[Vipul Naik]], _Topology: The journey into separation axioms_  [[[Naik-SeparationAxioms.pdf:file]]]

See also:

* Wikipedia, _[Separation axiom](http://secure.wikimedia.org/wikipedia/en/wiki/Separation_axiom)_

  (This is not really an independent reference, since one of the main authors of the present entry is also one of the main authors of the Wikipedia entry.)

Emphasis on the [[reflective subcategory|reflective]]-[[category theory]]:

* {#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))

[[!redirects separation axiom]]
[[!redirects separation axioms]]
[[!redirects separation property]]
[[!redirects separation properties]]

[[!redirects separation condition]]
[[!redirects separation conditions]]

[[!redirects topologically distinct point]]
[[!redirects topologically distinct points]]

[[!redirects topologically disjoint set]]
[[!redirects topologically disjoint sets]]
[[!redirects topologically disjoint subset]]
[[!redirects topologically disjoint subsets]]
[[!redirects topologically disjoint subspace]]
[[!redirects topologically disjoint subspaces]]

[[!redirects separated set]]
[[!redirects separated sets]]
[[!redirects separated subset]]
[[!redirects separated subsets]]
[[!redirects separated subspace]]
[[!redirects separated subspaces]]

[[!redirects perfectly normal topological space]]
[[!redirects perfectly normal topological spaces]]
[[!redirects perfectly normal space]]
[[!redirects perfectly normal spaces]]


separation axioms in terms of lifting properties

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

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

In [[point-set topology]], most of the [[separation axioms]] that are traditionally considered on [[topological spaces]] turn out  ([Gavrilovich 2014](#Gavrolovich14)) to have an equivalent reformulation in terms of [[lifting properties]], namely of the given space against, typically, a map of [[finite topological spaces]] which reflects the "opposite property" or the "archetypical counterexample" to the given separation condition, in a sense ("[[Quillen negation]]").

To see this, first notice/recall the following two basic examples of [[lifting properties]] in [[diagrams]] in [[TopSp]].

The property

* *$X \xrightarrow{f} Y$ is a [[surjective function]]*.

means equivalently that $f$ has the [[right lifting property]]
against the unique map from the [[empty space]] to the [[point space]],
the simplest map which is not surjective:

\begin{tikzcd}
[
  column sep={between origins, 40pt},
  row sep={between origins, 40pt}
]
  \varnothing
  \ar[rr]
  \ar[dd]
  &&
  X
  \ar[
    dd,
    "{ f }"
  ]
  \\
  \\
  \ast
  \ar[
    rr,
    "{ \forall }"
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  Y
\end{tikzcd}

Namely, a [[commuting diagram]] of the outer form is equivalently just the choice of a single point in $Y$ (this being the [[image]] of the bottom map), and the existence of the dashed lift means that any such point has a [[preimage]] through $f$. This is the very definition of [[surjective function|surjectivity]].

Note that we have defined surjectivity with help of a simplest counterexample: the map on the left is perhaps the simplest example
of a non-surjective map. We shall see that this pattern holds
in other examples as well. Part of the reason that a map never has the left lifting property with respect to itself unless it is an isomorphism,
and thus taking the lifting prperty is a simplest way to define a class of morphism not containing a given counterexample or without a given property
in a matter useful in a diagram chasing computation.

\linebreak

Similarly, the property

* *$X \xrightarrow{f} Y$ is an [[injective function]]*.

means equivalently that $f$ has the [[right lifting property]]
against the unique map from the [[discrete space]] with two elements $Dsc(\{0,1\})$

$$
  Opens
  \Big(
    Dsc
    \big(
      \{0,1\}
    \big)
  \Big)
  \;\;
  \coloneqq
  \;\;
  \Big\{
    \varnothing
    ,\,
    \{0\}
    ,\,
    \{1\}
    ,\,
    \{0,1\}
  \Big\}
$$

to the [[point space]], one of the simplest non-injective maps:

\begin{tikzcd}
[
  column sep={between origins, 40pt},
  row sep={between origins, 40pt}
]
  \mathrm{Dsc}
  \big(
    \boxed{\{\boxed{0},\boxed{1}\}}
  \big)
  \ar[
    rr,
    "{ \forall }"
  ]
  \ar[dd]
  &&
  X
  \ar[
    dd,
    "{ f }"
  ]
  \\
  \\
  \ast
  \ar[
    rr,
    "{ \forall }"
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  Y
\end{tikzcd}

Namely, that such a lift exists means that the two points in the [[image]] of the top map -- which may be any two points $x_1, x_2 \,\in\, X$ (by the nature of the [[discrete topology]]) such that (by [[commuting diagram|commutativity of the diagram]]) their images are equal, $f(x_1) \,=\, f(x_1) \;\in\; Y$ -- must already have been equal themselves. This is the very definition of [[injective function|injectivity]].

\linebreak

Now consider the variant of the previous example where the 2-element set is equipped instead with the [[codiscrete topology]], whose only [[open subsets]] are the [[empty set]] and the set containing both elements

\[
  \label{OpenSubsetsOfTheCodiscreteSpaceOnTwoElements}
  Opens
  \Big(
    CoDsc
   \big(
      \{0,1\}
   \big)
  \Big)
  \;\coloneqq\;
  \big\{
    \begin{array}{l}
      \varnothing, \{0,1\}
    \end{array}
  \big\}
  \,.
\]

In terms of this, the property

* *$X$ is a [[T0-space|$T_0$-space]].

means equivalently that its unique map $X \to \ast$ to the [[point space]] has the [[right lifting property]]
against the unique map from the [[codiscrete space]] with two [[elements]]
to the [[point space]]:

\begin{tikzcd}
[
  column sep={between origins, 40pt},
  row sep={between origins, 40pt}
]
  \mathrm{CoDsc}
  \big(
    \boxed{\{0\leftrightarrow 1\}}
  \big)
  \ar[
    rr,
    "{ \forall }"
  ]
  \ar[dd]
  &&
  X
  \ar[dd]
  \\
  \\
  \ast
  \ar[
    rr
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  \ast
\end{tikzcd}

Namely, now [[continuous function|continuity]] restricts the top map to be such that neither $x_i \in\; X$ is contained in an [[open subset]] that does not contain the other (this is the usual way to state the axiom): For if it were, then the [[pre-image]] of that open subset would be $\{x_i\} \,\subset\, CoDsc\big( \{0,1\} \big)$, which however is *not* open (eq:OpenSubsetsOfTheCodiscreteSpaceOnTwoElements). This means that if $X_1$ is a [[T0-space|$T_0$-space]] then any such top map must be the [[constant function]], and this is equivalent to the existence of a lift in the diagram.

Note that [[codiscrete space]] with two points is a simple example of not a $T_0$-space.

\linebreak

Proceeding in this manner, one sees that the property

* *$X$ is a [[T1-space|$T_1$-space]].

means equivalently that its unique map $X \to \ast$ to the [[point space]] has the [[right lifting property]]
against the unique map from the [[Sierpinski space]] $Sierp$,
again the simplest not a $T_1$-space.

$$
  Set
  \big(
    Sierp
  \big)
  \;\coloneqq\;
  \{0,1\}
  \,,
  \;\;\;\;\;\;
  Opens
  \big(
    Sierp
  \big)
  \;\coloneqq\;
  \big\{
    \varnothing
    ,\,
    \{0\}
    ,\,
    \{0,1\}
  \big\}
$$

to the [[point space]]:

\begin{tikzcd}
[
  column sep={between origins, 40pt},
  row sep={between origins, 40pt}
]
  \mathrm{\boxed{\{\boxed{0}\rightarrow 1\}}}
  \ar[
    rr,
    "{ \forall }"
  ]
  \ar[dd]
  &&
  X
  \ar[dd]
  \\
  \\
  \ast
  \ar[
    rr
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  \ast
\end{tikzcd}

Namely, the previous argument applies, but only to the point $1 \in Sierp$, while now $\{0\} \,\subset\, Sierp$ *is* [[open subset|open]]. Therefore the existence of lifts now means that any two points must *both* have an [[open neighbourhood]] not containing the other, which is the definition of a [[T1-space|$T_1$-space]].

Next, a space is [[Hausdorff space|Hausdorff]] (axiom $T_2$) iff every two distinct points have disjoint open neighbourhoods. To give two disjoint open subsets is to give a map to the space with two open points and one closed point. To give two distinct points is to give an injective map. Hence, every two
distinct points have disjoint open neighbourhoods iff any such injective map extends to a map to that space with three points.

* *$X$ is a [[Hausdorff space|$T_2$-space]].

{#IllustrationOfLiftingForT2} This is represented by the following lifting property diagram.

\begin{tikzcd}
[
  column sep={between origins, 50pt},
  row sep={between origins, 40pt}
]
  \boxed{\{0\leftrightarrow 1\}}
  \ar[
    rr,
    "{ }"
  ]
  \ar[dd, hook, "{ \forall  }"{left} ]
  &&
  \boxed{
    \overset{
      \boxed{
        \boxed{u}
        \;\;
        \,
        \;\;
        \boxed{v}}
      }{
        \underset{
           c
        }
        {
          \searrow \;\, \swarrow
        }
      }
  }
  \ar[dd]
  \\
  \\
  X
  \ar[
    rr
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  \ast
\end{tikzcd}

\linebreak

Axioms $T_3-T_5$ and others require finite topological spaces with 4 to 7 points, and we need to introduce appropriate notation.

We give one more reformulation which does not require special notation.
A space is [[extremally disconnected]] iff the closure of an open subset is open, or, equivalently, the closures of disjoint open subsets are disjoint.

* *$X$ is [[extremally disconnected]].

{#IllustrationOfLiftingForExtremallyDisconnected} This is represented by the following lifting property diagram.

\begin{tikzcd}
[
  column sep={between origins, 50pt},
  row sep={between origins, 40pt}
]
 \emptyset
  \ar[
    rr,
    "{ }"
  ]
  \ar[dd]
  && \boxed{
\boxed{{}^{\boxed{u}}\!\! \searrow_{\,{}_{c_1}}} \,\,\, \boxed{{}_{{}_{c_2}}\swarrow^{\boxed{v}}}
}
   \ar[dd]
  \\
  \\
  X
  \ar[
    rr,
    "{ \forall  }"{below}
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  \boxed{
    \overset{
      \boxed{
        \boxed{u}
        \;\;
        \,
        \;\;
        \boxed{v}}
      }{
        \underset{
           c
        }
        {
          \searrow \;\, \swarrow
        }
      }
  }
\end{tikzcd}

\linebreak

## Background and notation
 {#BackgroundAndNotation}

In order to economically define and denote the [[finite topological spaces]] which will appear in the [[lifting problems]] discussed below, we may encode them through their *[[specialization order]]*.

Recall that the [[specialisation preorder]] on the [[underlying]] [[set]] of points of a [[topological space]] $X$ is the [[preorder]] whose [[order relation]], for any $x, y \in X$, is

\[
  \label{SpecializationPreorder}
  x \,\leq\, y
  \;\;\;\;\;\;\text{iff}\;\;\;\;\;\;
  x \,\in\, cl(y)
  \,,
\]

where the right hand side means that the following two equivalent conditions hold:

1. $x$ is in the [[topological closure]] of $y$;

1. every [[open subset]] which contains $x$ also contains $y$.

{#ProsetsAsCategories} We may regard these [[preordered sets]] equivalently as ([[thin category|thin]] and [[strict category|strict]]) [[categories]], whose

* [[objects]] are the points of $X$,

* [[morphisms]] reflect the order relation:

  for $x, y \,\in\, X$ there exists a unique morphism

  $$
    y \,\leftarrow\, x
    \;\;\;\text{iff}\;\;\;
    y \,\leq\, x\;
  $$

  (the mnemonic: view the arrow head of "$\leftarrow$" as a "$\lt$"-symbol)

For a [[finite topological space]] $X$, this [[specialisation preorder]] $Spec(X)$ -- or equivalently the corresponding category, which we shall conceptually conflate with the pre-ordering -- uniquely determines the topology:

A [[subset]] $C \,\subset X\,$ is [[closed subset|closed]] iff the following equivalent conditions hold:

1. $C$ [[downward closed subset|downward closed]] in the specialization order;

1. there are no morphisms going out of $C$ in the corresponding category.

Accordingly, we may and will denote [[finite topological spaces]] by showing the [[graph]] containing their points with a system of arrows indicating the generating [[morphisms]] in their corresponding [[specialization preorder]]-[[thin category|category]].

In doing so, often it will be convenient to show multiple copies of the *same* object, i.e. the same point. Noticing that in the [[strict category]] corresponding to a [[preorder]], an [[isomorphism]] between two objects does *not imply* their [[equality]], we have and distinguish the following two notations:

* '$=$' denotes an [[identity morphism]],

* '$\leftrightarrow$' denotes an [[isomorphism]]

  (hence, because the [[category]] is [[thin category|thin]], the existence of arrows back and forth $\leftrightarrows$).

{#BasicExamplesForNotation} For example:

|[[finite topological space]] |[[open subsets]] |[[specialization order]]|as picture |
|--|--|--|--|
| [[discrete space]]
  $Dsc\big(\{ 0,1 \}\big)$ | $\Big\{\; \varnothing,\, \{0\},\, \{1\},\, \{0,1\} \;\Big\}$ | $\Big\{\; 0 \phantom{\leftarrow} 1 \;\Big\}$| $\boxed{\{\boxed{0},\boxed{1}\}}$ |
| [[Sierpinski space]]
 $Sierp$ | $\Big\{\; \varnothing,\, \{0\},\, \{0,1\} \;\Big\}$ | $\Big\{\; 0 \rightarrow 1 \;\Big\}$ | $\boxed{\{\boxed{0}\rightarrow 1\}}$|
| [[codiscrete space]]
 $CoDsc\big( \{0,1\} \big)$ | $\Big\{\; \varnothing,\, \{0,1\}  \;\Big\}$ | $\Big\{\; 0 \leftrightarrow 1 \;\Big\}$ |  $\boxed{\{0\leftrightarrow 1\}}$ |
| [[point space]]
 $\ast$ | $\Big\{ \varnothing,\, \{0\} = \{1\}  \;\Big\}$ |  $\Big\{\; 0 = 1 \;\Big\}$  | $\boxed{*}$ |

{#NoArrowsFromClosedPoints} Notice here how in $\big\{\; 0 \rightarrow 1 \;\}$ the point $0$ is not [[closed point|closed]] (as there do emanate arrows form it) while the point ${1}$ is [[closed point|closed]] (as no arrows emanate from it).

Under this identitification of [[finite topological spaces]] $X$ with  [[preordered sets]] regarded as [[thin categories]] $Spec(X)$, the [[continuous maps]] between topological spaces correspond to [[functors]] between their specialization preorders:

$$
  \overset{
    \color{blue}
    {
      \text{continuous function betw.}
      \atop
      \text{finite topological spaces}
    }
    \mathclap{\phantom{\vert_{\vert_{\vert_{\vert_{\vert_{\vert}}}}}}}
  }{
    X \xrightarrow{f} Y
  }
  \;\;\;\;\;\;\;\;\;
  \text{corresponds to}
  \;\;\;\;\;\;\;\;\;
  \overset{
    \color{blue}
    {
      \text{functor between}
      \atop
      \text{ specialization orders }
    }
    \mathclap{\phantom{\vert_{\vert_{\vert_{\vert_{\vert_{\vert}}}}}}}
  }{
    Spec(X) \xrightarrow{Spec(f)} Spec(Y)
  }
  \mathrlap{\,.}
$$

With specialization orders denoted by their generating graphs as before, and using that there is at most one morphism for every ordered pair of objects, we may specify such functors $Spec(f)$ simply by labeling each object in their [[codomain]] by the same symbol as its [[preimage]].

For example

$$
  \big\{\, 0 \,\big\}
  \;\;\;
  \xrightarrow{\phantom{---}}
  \;\;\;
  \big\{\, 0,\, 1  \,\big\}
$$

is to denote the [[functor]] between [[specialization orders]] of [[discrete spaces]] with a single and with two elements, respectively, which takes the point denoted "$0$" on the left to the point denoted by the same symbol "$0$" on the right.

{#NotationForFiniteTopologicalSpaces}
In this notation, the following shows the canonical functors between the four examples of specialization orders from the above list:

$$
  \overset{
    \color{blue}
    {
      \text{discrete space}
      \atop
      \phantom{-}
    }
  }{
    \Big\{\;
      0
      \phantom{\leftarrow}
      1
    \;\Big\}
  }
  \;\;\xrightarrow{\phantom{---}}\;\;
  \overset{
    \color{blue}
    {
      \text{Sierpinski space}
      \atop
      \phantom{-}
    }
  }{
    \Big\{\;
      0
      \rightarrow
      1
    \;\Big\}
  }
  \;\;\xrightarrow{\phantom{---}}\;\;
  \overset{
    \color{blue}
    {
      \text{codiscrete space}
      \atop
      \phantom{-}
    }
  }{
    \Big\{\;
      0
      \leftrightarrow
      1
    \;\Big\}
  }
  \;\;\xrightarrow{\phantom{---}}\;\;
  \overset{
    \color{blue}
    {
      \text{point space}
      \atop
      \phantom{-}
    }
  }{
    \Big\{\;
      0
      =
      1
    \;\Big\}
  }
  \mathrlap{\,.}
$$

{#ExampleMapIdentifyingThreePoints} Notice here the role of the equality sign: In the denotation of a  functor as above, arrows may be sent to equality signs (but not the other way around): This corresponds to the corresponding [[continuous function]] "gluing" these points, in that it is (at least locally) the [[coprojection]] onto the [[quotient space|quotient]] by the subset of points that are being identified.

For example, the following denotes the functor corresponding to a map that "glues" three points to each other:

$$
  \left\{
    \;\;
    \array{
      & & U && && V
      \\
      & \swarrow && \searrow && \swarrow && \searrow
      \\
      a && && x && && b
    }
    \;\;
  \right\}
  \;\;\;\;\;\;\;\;
  \xrightarrow{\phantom{------}}
  \;\;\;\;\;\;\;\;
  \left\{
    \;\;
     \array{
       && U  &\color{red}=& x &\color{red}=& V
       \\
       & \swarrow & && && & \searrow
       \\
       a & & &&  &&  &&   b
     }
    \;\;
  \right\}
$$
In this notation, the open subsets of the domain are
$$\{\{U\}, \{V\}, \{a,U\}, \{V,b\}, \{U,x,V\}\},\{a,U,x,V,b\}\}
$$
In the codomain, points $a$ and $b$ are closed, and the only point left is open.

## Separation of subsets
 {#SeparationOfSubsets}

Here we describe how topological separation of *[[subsets]]* of a [[topological space]] may be expressed in terms of factorizations of their joint [[characteristic function]].

In all of the following, $S$ denotes a [[topological space]] and

$$
  F,\,G \;\subset\; S
$$

a [[pair]] of [[subsets]].

### Disjoint

We say that a pair of subsets is [[disjoint subset|disjoint]] if their [[intersection]] is [[empty set|empty]]:

\[
  \label{APairOfDisjointSubsets}
  F,\, G \,\subset\, S
  \,,
  \;\;\;\;
  F \cap G \,=\, \varnothing
\]

This situation (eq:APairOfDisjointSubsets) may equivalently be expressed by a *[[characteristic function|characteristic]]* [[continuous function]] from $S$ to the [[codiscrete topological space]] of 3 elements. We will suggestively denote these three elements by $e_F$, $e_G$ and $e_\varnothing$, respectively, so that in the [above](#BackgroundAndNotation) notation the codiscrete topology on them reads like this:

$$
  CoDisc
  \big(
    \{ e_F ,\, e_F ,\, e_{\varnothing} \}
  \big)
  \;\;
  =
  \;\;
  \big\{
    e_F \leftrightarrow e_G \leftrightarrow e_\varnothing
  \big\}
  \,.
$$

Namely, any function to such a [[codiscrete space]] is continuous, and any function to the set with 3 elements partitions its [[domain]] into the [[disjoint]] [[preimages]] of these three elements, which we may regard as a pair $F, G$ of [[disjoint subsets]] and their [[complement]] $S \setminus \{F \cup G\}$:

\[
  \label{CharacteristicFunctionForPairOfDIsjointSubsets}
  \array{
    S_{F,G}
    &\colon&
    S
    &\longrightarrow&
    \big\{
       e_F \leftrightarrow e_G \leftrightarrow e_{\varnothing}
    \big\}
    \\
    &&
    x
    &\mapsto&
    \left\{
    \array{
      e_F &\vert& x \in F
      \\
      e_G &\vert& x \in G
      \\
      e_\varnothing &\vert& otherwise
    }
    \right.
  }
\]

We may now encode topological separation properties of the two subsets in terms of factorizations (hence liftings) of this their [[characteristic function]]:

Often $F$ and $G$ will be points (identified with their [[singleton]] subsets); in that case, one usually says _distinct_ in place of _disjoint_.

Often $F$ or $G$ will be closed sets; notice that disjoint closed sets are automatically separated, while a closed set and a point, if disjoint, are automatically topologically disjoint.

To express the assumption that $F$ or $G$ is closed, we may modify the topology on $\{e_F,e_{\varnothing}, e_G$ and make the points $e_F$ and/or $e_G$ closed as appropriate.

### Topologically disjoint
 {#TopologicallyDisjoint}

We say that a disjoint pair of subsets (eq:APairOfDisjointSubsets) is __topologically disjoint__ if there exists a [[neighbourhood]] of one set that is [[disjoint subset|disjoint]] from the other set:

\[
  \label{PairOfTopologicallyDisjointSubsets}
     F, G \;\text{are topologically disjoint}
     \;\;\;\;\;\;\;\;\;
     \Leftrightarrow
     \;\;\;\;\;\;\;\;\;
     \Big(
       \underset
         {
           U \underset{\mathclap{nbhd}}{\supseteq} F
         }
         {\exists}
         \;\;
       U \cap G = \varnothing
     \Big)
     \;\;or\;\;
     \Big(
       \underset
         {
           V \underset{\mathclap{nbhd}}{\supseteq} G
         }
         {\exists}
       \;\;
       F \cap V = \varnothing
     \Big)
     \,.
\]

(Notice that topologically disjoint sets must be disjoint.)

The topological separation condition (eq:PairOfTopologicallyDisjointSubsets) on a pair of disjoint subsets means equivalently that their [[characteristic function]] (eq:CharacteristicFunctionForPairOfDIsjointSubsets) factors as

\begin{tikzcd}
[
  column sep={between origins, 60pt},
  row sep={between origins, 40pt}
]
  \varnothing
  \ar[dd]
  \ar[rr]
  &&
  \big\{
    e_F
      \leftrightarrow
    e_U
      \to
    e_{\varnothing}
      \leftrightarrow
    e_G
  \big\}
  \ar[dd]
  \\
  \\
  S
  \ar[
    rr,
    "{ S_{F,G} }"{below}
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  \big\{
    e_F
      \leftrightarrow
    e_U
      =
    e_{\varnothing}
      \leftrightarrow
    e_G
  \big\}
\end{tikzcd}

or as

\begin{tikzcd}
[
  column sep={between origins, 60pt},
  row sep={between origins, 40pt}
]
  \varnothing
  \ar[dd]
  \ar[rr]
  &&
  \big\{
    e_F
      \leftrightarrow
    e_{\varnothing}
      \leftarrow
    e_V
      \leftrightarrow
    e_G
  \big\}
  \ar[dd]
  \\
  \\
  S
  \ar[
    rr,
    "{ S_{F,G} }"{below}
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  \big\{
    e_F
      \leftrightarrow
    e_{\varnothing}
      =
    e_U
      \leftrightarrow
    e_G
  \big\}
\end{tikzcd}

The required neighbourhoods $U\supset F$ or $V\supset B$ are
the preimages under the diagonal arrow of the open neighbourhoods
$ \big\{
    e_F
      \leftrightarrow
        e_{\varnothing}
  \big\}$ of point $e_F$, or
$ \big\{
        e_{\varnothing}
      \leftrightarrow
    e_G
  \big\}
$ of point $e_G$, respectively.

### Topologically separated
 {#TopologicallySeparated}

We say that a disjoint pair of subsets (eq:APairOfDisjointSubsets) is
__separated__ if each set has a [[neighbourhood]] that is [[disjoint subset|disjoint]] from the other set:

\[
  \label{PairOfSeparatedSubsets}
     F, G \;\text{are topologically disjoint}
     \;\;\;\;\;\;\;\;\;
     \Leftrightarrow
     \;\;\;\;\;\;\;\;\;
     \Big(
       \underset
         {
           U \underset{\mathclap{nbhd}}{\supseteq} F
         }
         {\exists}
       ,\;
       U \cap G = \varnothing
     \Big)
     \;\;and\;\;
     \Big(
       \underset
         {
           V \underset{\mathclap{nbhd}}{\supseteq} G
         }
         {\exists}
       ,\;
       F \cap V = \varnothing
     \Big)
     \,.
\]

(Notice that separated sets must be topologically disjoint and disjoint.)

The separation condition (eq:PairOfSeparatedSubsets) on a pair of disjoint subsets means equivalently that
their characteristic function
(eq:CharacteristicFunctionForPairOfDIsjointSubsets)
factors as

\begin{tikzcd}
[
  column sep={between origins, 60pt},
  row sep={between origins, 40pt}
]
  \varnothing
  \ar[dd]
  \ar[rr]
  &&
  \big\{
    e_F
      \leftrightarrow
    e_U
      \to
    e_{\varnothing}
      \leftrightarrow
    e_G
  \big\}
  \ar[dd]
  \\
  \\
  S
  \ar[
    rr,
    "{ S_{F,G} }"{below}
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  \big\{
    e_F
      \leftrightarrow
    e_U
      =
    e_{\varnothing}
      \leftrightarrow
    e_G
  \big\}
\end{tikzcd}

and as

\begin{tikzcd}
[
  column sep={between origins, 60pt},
  row sep={between origins, 40pt}
]
  \varnothing
  \ar[dd]
  \ar[rr]
  &&
  \big\{
    e_F
      \leftrightarrow
    e_{\varnothing}
      \leftarrow
    e_V
      \leftrightarrow
    e_G
  \big\}
  \ar[dd]
  \\
  \\
  S
  \ar[
    rr,
    "{ S_{F,G} }"{below}
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  \big\{
    e_F
      \leftrightarrow
    e_{\varnothing}
      =
    e_V
      \leftrightarrow
    e_G
  \big\}
\end{tikzcd}

The two diagrams above can equivalently be combined into a single diagram;
the separating neighbourhoods $U\supset F$ and $V\supset B$ are
the preimages under the diagonal arrow of the open subsets
$ \big\{
    e_F
      \leftrightarrow
    e_U \leftarrow
        e_{\varnothing}
  \big\}$ and
$ \big\{
        e_{\varnothing}
      \rightarrow
    e_V
      \leftrightarrow
    e_G
  \big\}
$.

\begin{tikzcd}
[
  column sep={between origins, 60pt},
  row sep={between origins, 40pt}
]
  \varnothing
  \ar[dd]
  \ar[rr]
  &&
  \big\{
    e_F
      \leftrightarrow
    e_U \leftarrow
        e_{\varnothing}
      \rightarrow
    e_V
      \leftrightarrow
    e_G
  \big\}
  \ar[dd]
  \\
  \\
  S
  \ar[
    rr,
    "{ S_{F,G} }"{below}
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  \big\{
    e_F
      \leftrightarrow
    e_U=
    e_{\varnothing}
      =
    e_V
      \leftrightarrow
    e_G
  \big\}
\end{tikzcd}

### Separated by neighbourhoods
 {#SeparatedByNeighbourhoods}

We say that a disjoint pair of subsets (eq:APairOfDisjointSubsets) is
__separated by neighbourhoods__ if the sets have [[disjoint subset|disjoint]] [[neighbourhoods]]

\[
  \label{PairOfSeparatedNeighbourhoods}
  U \subseteq V
  \;\text{and}\;
  V \subseteq G
  \;\;\;\text{such that}\;\;\;
  U \cap V = \varnothing
  \,.
\]

The separation condition (eq:PairOfSeparatedNeighbourhoods) on a pair of disjoint subsets means equivalently that
their characteristic function
(eq:CharacteristicFunctionForPairOfDIsjointSubsets)
factors as

\begin{tikzcd}
[
  column sep={between origins, 60pt},
  row sep={between origins, 40pt}
]
  \varnothing
  \ar[dd]
  \ar[rr]
  &&
  \big\{
    e_F
      \leftrightarrow
    e_U \rightarrow
        e_{\varnothing}
      \leftarrow
    e_V
      \leftrightarrow
    e_G
  \big\}
  \ar[dd]
  \\
  \\
  S
  \ar[
    rr,
    "{ S_{F,G} }"{below}
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  \big\{
    e_F
      \leftrightarrow
    e_U=
    e_{\varnothing}
      =
    e_U
      \leftrightarrow
    e_G
  \big\}
\end{tikzcd}
The separating neighbourhoods $U\supset F$ and $V\supset B$ are
the preimages under the diagonal arrow of the open subsets
$ \big\{
    e_F
      \leftrightarrow
    e_U
  \big\}$ and
$ \big\{e_V
      \leftrightarrow
    e_G
  \big\}
$.

### Separated by closed neighbourhoods

{#eq:PairOfSeparatedClosedNeighbourhoods}
We say that a disjoint pair of subsets (eq:APairOfDisjointSubsets) is
__separated by neighbourhoods__ if the sets have [[disjoint subset|disjoint]] closed [[neighbourhoods]], i.e. there exist
$U\subseteq V$ and $V\subseteq G$ such that their closures
$\bar U$ and $ \bar V$ do not intersect

\[
  \label{PairOfSeparatedClosedNeighbourhoods}
  U \subseteq V
  \;\text{and}\;
  V \subseteq G
  \;\;\;\;\;\;
  \text{such that}
  \;\;\;\;\;\;
  \bar U \cap \bar V
  \;=\;
  \varnothing
\]

The separation condition (eq:PairOfSeparatedClosedNeighbourhoods) on a pair of disjoint subsets means equivalently that
their characteristic function
(eq:CharacteristicFunctionForPairOfDIsjointSubsets)
factors as

\begin{tikzcd}
[
  column sep={between origins, 75pt},
  row sep={between origins, 40pt}
]
  \varnothing
  \ar[dd]
  \ar[rr]
  &&
  \big\{
    e_F
      \leftrightarrow
    e_U \rightarrow
    e_{\bar U} \leftarrow
        e_{\varnothing} \rightarrow
    e_{\bar V}
      \leftarrow
    e_V
      \leftrightarrow
    e_G
  \big\}
  \ar[dd]
  \\
  \\
  S
  \ar[
    rr,
    "{ S_{F,G} }"{below}
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  \big\{
    e_F
      \leftrightarrow
    e_U=e_{\bar U}=
    e_{\varnothing}
      =
    e_{\bar V}=e_V
      \leftrightarrow
    e_G
  \big\}
\end{tikzcd}

The separating closed neighbourhoods $\bar U\supset F$ and $\bar V\supset B$ are
the preimages under the diagonal arrow of the closed neighbourhoods
$ \big\{
    e_F
      \leftrightarrow
   e_U \rightarrow
   e_{\bar U} \big\}$ of point $e_F$, and
$ \big\{
  e_{\bar V}
      \leftarrow
  e_V
      \leftrightarrow
    e_G
  \big\}
$ of point $e_G$.

### Separated by a function

We say that two disjoint subsets $F$ and $G$ are __separated by a function__ if there exists a continuous [[real number|real]]-valued [[function]] on the space that maps $F$ to $0$ and $G$ to $1$:

\[
  \label{ConditionForSeparationByAFunction}
  \underset
    {f \,\colon\, S \to \mathbb{R}}
    {\exists}
  \left(
    F \subseteq f^{-1}(\{0\})
    \;\wedge\;
    G \subseteq f^{-1}(\{1\})
  \right)
\]

Equivalently, we may assume that $f$ takes values in $[0,1]\subseteq \mathbb{R}$.

This separation condition (eq:ConditionForSeparationByAFunction) on a pair of disjoint subsets means equivalently that
their characteristic function
(eq:CharacteristicFunctionForPairOfDIsjointSubsets)
factors as

\begin{tikzcd}
[
  column sep={between origins, 60pt},
  row sep={between origins, 40pt}
]
  \varnothing
  \ar[dd]
  \ar[rr]
  &&
  {[0,1]}\vee_{\{0,1\}} \{e_F\leftrightarrow 0, 1\leftrightarrow e_G\}
  \ar[dd]
  \\
  \\
  S
  \ar[
    rr,
    "{ S_{F,G} }"{below}
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  \big\{
    e_F
      \leftrightarrow
    e_{[0,1]}
      \leftrightarrow
    e_G
  \big\}
\end{tikzcd}
If $f:S\to [0,1]$ in a separating function as above, we may take the diagonal arrow defined as
\[
  \label{CharacteristicFunctiontoR01ForPairOfDIsjointSubsets}
  \array{
    f'
    &\colon&
    S
    &\longrightarrow& [0,1]\vee_{\{0,1\}} \{0'\leftrightarrow 0, 1\leftrightarrow 1\}
    \\
    &&
    x
    &\mapsto&
    \left\{
    \array{
      e_F &\vert& x \in F
      \\
      e_G &\vert& x \in G
      \\
      f(x) &\vert& otherwise
    }
    \right.
  }
\]

### Precisely separated by a function

Finally, we say that two disjoint subsets $F$ and $G$ are __separated by a function__ if there exists a continuous [[real number|real]]-valued [[function]] on the space that maps precisely $F$ to $0$ and $G$ to $1$:

\[
  \label{ConditionForPreciseSeparationByAFunction}
  \underset
    {f \,\colon\, S \to \mathbf{R}}
    {\exists}
  \left(
    F \,=\, f^{-1}(\{0\})
    \;\;\wedge\;\;
    G \,=\, f^{-1}(\{1\})
  \right)
\]

Equivalently, we may assume that $f$ takes values in $[0,1]\subseteq \mathbb{R}$.

This separation condition (eq:ConditionForPreciseSeparationByAFunction) on a pair of disjoint subsets means equivalently that
their characteristic function
(eq:CharacteristicFunctionForPairOfDIsjointSubsets)
factors as

\begin{tikzcd}
[
  column sep={between origins, 60pt},
  row sep={between origins, 40pt}
]
  \varnothing
  \ar[dd]
  \ar[rr]
  && [0,1]
  \ar[dd]
  \\
  \\
  S
  \ar[
    rr,
    "{ S_{F,G} }"{below}
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  \big\{
    0
      \leftrightarrow
    e_{(0,1)}
      \leftrightarrow
    1
  \big\}
\end{tikzcd}
If $f:S\to [0,1]$ in a separating function as above, we may take the diagonal arrow defined as
\[
  \label{CharacteristicFunctiontoRForPairOfDIsjointSubsets}
  \array{
    f'
    &\colon&
    S
    &\longrightarrow&
  {[0,1]}
    \\
    &&
    x
    &\mapsto&
    \left\{
    \array{
      0 &\vert& x \in F
      \\
      1 &\vert& x \in G
      \\
      f(x) &\vert& otherwise
    }
    \right.
  }
\]

  Notice that sets separated by a function must be separated by closed neighbourhoods (the preimages of $[-\epsilon, \epsilon]$ and $[1-\epsilon, 1+\epsilon]$).
   Notice that sets precisely separated by a function must be separated by a function.

\linebreak

## Separation axioms as lifting properties
 {#SeparationAxiomsAsLiftingProperties}

In all of the following definitions, ${X}$ is  a topological space.

We shall use the symbol "$⧄$" to denote the [[lifting property]] of one map against another (compare *[[Joyal-Tierney calculus]]*):

So for $S \xrightarrow{\phi} T $ and $X \xrightarrow{f} Y$ a [[pair]] of [[continuous functions]] between topological spaces, we write

$$
  \phi \;⧄\; f
  \;\;\;\;\;\;
  \Leftrightarrow
  \;\;\;\;\;\;
  \phi \;\text{has the left lifting property against}\; f
$$

\begin{tikzcd}
[
  column sep={between origins, 40pt},
  row sep={between origins, 40pt}
]
  S
  \ar[
    rr,
    "{ \forall }"
  ]
  \ar[
    dd,
    "{ \phi }"
  ]
  &&
  X
  \ar[
    dd,
    "{ f }"
  ]
  \\
  \\
  T
  \ar[
    rr,
   "{ \forall }"
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  Y
\end{tikzcd}

With this notation, we have the following dictionary between [[lifting properties]] and [[separation axioms]].

### Kolmogorov spaces ($T_0$)
 {#KolmogorovSpaces}

A topological space ${X}$  is [[T0-space|$T_0$]] (or [[Kolmogorov topological space|Kolmogorov]]) if any two distinct points in ${X}$ are [topologically disjoint](#TopologicallyDisjoint).

(It will be a common theme among the following separation axioms to have
one version of an axiom that requires $T_0$ and one version that doesn't.)

This condition is equivalent to the [[lifting property]]

$$
  \big(
    \{x\leftrightarrow y\} \longrightarrow  \{x=y\}
  \big)
  \;\;\,⧄\,\;\;
  \big(
    {X} \longrightarrow  \{*\}
  \big)
$$

hence:

\begin{tikzcd}
[
  column sep={between origins, 40pt},
  row sep={between origins, 40pt}
]
  \mathrm{CoDsc}
  \big(
    \boxed{\{0\leftrightarrow 1\}}
  \big)
  \ar[
    rr,
    "{ \forall }"
  ]
  \ar[dd]
  &&
  X
  \ar[dd]
  \\
  \\
  \ast
  \ar[
    rr
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  \ast
\end{tikzcd}

The same property is also defined by the following diagram:

\begin{tikzcd}
[
  column sep={between origins, 40pt},
  row sep={between origins, 40pt}
]
    \{\boxed{0}\}
  \ar[
    rr,
    "{ \forall }"
  ]
  \ar[dd]
  &&
  X
  \ar[dd]
  \\
  \\
  \mathrm{CoDsc}
  \big(
    \boxed{\{0\leftrightarrow 1\}}
  \big)
  \ar[
    rr]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  \ast
\end{tikzcd}

### Symmetric spaces ($R_0$)
 {#SymmetricSpaces}

A topological space ${X}$ is an [[R0-space|$R_0$-space]] (also: *symmetric space*), if any two [topologically distinguishable](#TopologicallyDisjoint) points in $X$ are [topologically separated](#TopologicallySeparated).

This condition is equivalent to the following [[lifting property]]:

$$
  \big(
    \{x{\searrow}y\} \longrightarrow  \{x\leftrightarrow y\}
  \big)
  \;\;\;\,⧄\,\;\;\;
  \big(
    {X} \longrightarrow  \{*\}
  \big)
  \,.
$$

### Accessible spaces ($T_1$)
 {#LiftingPropertyCorrespondingtoT1}

$X$ is [[T1-space|$T_1$]] (or *accessible* or *Fréchet*) if any two distinct points in $X$ are separated, i.e.

   $$
     \{x{\searrow}y\} \longrightarrow  \{x=y\}
     \;\;\;\,⧄\,\;\;\;
     {X} \longrightarrow  \{*\}
   $$

 Thus, $X$ is T1 if and only if it is both T0 and R0: indeed, a morphism lifts wrt the composition
$ \{x{\searrow}y\} \longrightarrow  \{x\leftrightarrow y\} \longrightarrow  \{x=y\}$
iff it lifts wrt either one of the two morphisms. (Although you may
say such things as "T1 space", "Frechet topology", and "Suppose that the
topological space $X$ is Frechet", avoid saying "Frechet space" in this
context, since there is another entirely different notion of Frechet space in
functional analysis.)

\begin{tikzcd}
[
  column sep={between origins, 40pt},
  row sep={between origins, 40pt}
]
 \{\boxed{\boxed{0}\rightarrow 1}\}
  \ar[
    rr,
    "{ \forall }"
  ]
  \ar[dd]
  &&
  X
  \ar[dd]
  \\
  \\
  \ast
  \ar[
    rr
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  \ast
\end{tikzcd}

The same property is also defined by the following diagrams:

\begin{tikzcd}
[
  column sep={between origins, 40pt},
  row sep={between origins, 40pt}
]
    \{\boxed{0}\}
  \ar[
    rr,
    "{ \forall }"
  ]
  \ar[dd]
  &&
  X
  \ar[dd]
  \\
  \\
 \{\boxed{\boxed{0}\rightarrow 1}\}
  \ar[
    rr]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  \ast
\end{tikzcd}

\begin{tikzcd}
[
  column sep={between origins, 40pt},
  row sep={between origins, 40pt}
]
    \{\boxed{1}\}
  \ar[
    rr,
    "{ \forall }"
  ]
  \ar[dd]
  &&
  X
  \ar[dd]
  \\
  \\
 \{\boxed{\boxed{0}\rightarrow 1}\}
  \ar[
    rr]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  \ast
\end{tikzcd}

### Reregular spaces ($R_1$)

$X$ is R1, or preregular, if any two topologically distinguishable points in X are separated by neighbourhoods. Every R1 space is also R0.

### Weakly Hausdorff spaces

$X$ is [[weakly Hausdorff topological space|weakly Hausdorff]], if the image of every continuous map from a [[compact Hausdorff space]] into $X$ is closed. All weak Hausdorff spaces are T1, and all Hausdorff spaces are weak Hausdorff.

### Hausdorff spaces ($T_2$)

$X$ is [[Hausdorff space|Hausdorff]], or T2 or separated, if any two distinct points in $X$ are separated by neighbourhoods, i.e.

   $$
     \{x,y\} \hookrightarrow  \underset{X}{}
     \;\;\;\,⧄\,\;\;\;
       \{x{\searrow}\underset{X}{}{\swarrow}y\} \longrightarrow  \{x=X=y\}
   $$
As a lifting diagram this is
\begin{tikzcd}
[
  column sep={between origins, 80pt},
  row sep={between origins, 40pt}
]
  \{\boxed{x}\leftrightarrow \boxed{y}\}
  \ar[
    rr,
    "{ }"
  ]
  \ar[dd, hook, "{ \forall  }"{left} ]
  &&
  \boxed{
    \overset{
      \boxed{
        \boxed{x}
        \;\;
        \,
        \;\;
        \boxed{y}}
      }{
        \underset{
           X
        }
        {
          \searrow \;\, \swarrow
        }
      }
  }
  \ar[dd]
  \\
  \\
  X
  \ar[
    rr
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
   \{x=X=y\}
\end{tikzcd}
The interesting case is when the top horizontal arrow
$     \{x,y\} \to
       \{x{\searrow}\underset{X}{}{\swarrow}y\} $
maps $x$ and $y$ to the open points $x$ and $y$;
in that case the separating neighbourhoods are the preimages of the open points $x$ and $y$ under the diagonal arrow.
In the other cases both points map to a copy of $\{0\to 1\}$,
and the lifting property reduces to $T_1$. In particular,
we see that every  Hausdorff space is also T1.

   Also, $X$ is Hausdorff if and only if it is both T0 and R1.

### Urysohn spaces ($T_{2 \tfrac{1}{2}}$)

$X$ is $T2\frac{1}{2}$, or Urysohn, if any two distinct points in $X$ are separated by closed neighbourhoods. As a lifting property this is
   $$
    \{x,y\} \hookrightarrow  {X}
    \;\;\;\,⧄\,\;\;\;
     \boxed{\{\overset{\boxed{x}}{}{\searrow}\underset{U}{}{\swarrow}\overset{\boxed{X}}{}
}\!\!\!\!\!\!\!
\boxed{
{\,\,\,\,\,\,\searrow}\underset{V}{}{\swarrow}\overset{\boxed{y}}{}
}
\}     \longrightarrow  \{\boxed{x=x'=X=y'=y}\}
   $$

As a lifting diagram this is
\begin{tikzcd}
[
  column sep={between origins, 70pt},
  row sep={between origins, 40pt}
]
  \{\boxed{0}\leftrightarrow \boxed{1}\}
  \ar[
    rr,
    "{ }"
  ]
  \ar[dd, hook, "{ \forall  }"{left} ]
  &&
 \{\boxed{\overset{\boxed{x}}{}{\searrow}\underset{U}{}{\swarrow}\overset{\boxed{X}}{}
}\!\!\!\!\!\!\!\!\!\!\!
\boxed{
{\,\,\,\,\,\,\,\,\,\,\searrow}\underset{V}{}{\swarrow}\overset{\boxed{y}}{}
}
\}
  \ar[dd]
  \\
  \\
  X
  \ar[
    rr
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
\{\boxed{x=x'=X=y'=y}\}
\end{tikzcd}
The interesting case is when the top horizontal arrow
maps $x$ and $y$ to the open points $x$ and $y$;
in that case the separating closed neighbourhoods are
of closed subsets
$\big\{\overset{\boxed{x}}{}{\searrow}\underset{U}{}\big\} $
and $\big\{\underset{V}{}{\swarrow}\overset{\boxed{y}}{}\big\}$
under the diagonal arrow. The other cases are similar but easier.

Note the 5-point space here contains a copy of the 3-point space used in $T_2$. Hence,    every $T2\frac{1}{2}$ space is also Hausdorff.

### Completely Hausdorff spaces

$X$ is completely Hausdorff, or completely T2, if any two distinct points in X are separated by a continuous function, i.e.

   $$
     \{x,y\} \hookrightarrow  {X}
     \;\;\;\,⧄\,\;\;\;
     [0,1]\longrightarrow \{*\}
   $$

   where  $\{x,y\} \hookrightarrow  X$  runs through all injective maps from the discrete two point space $\{x,y\}$.
As a lifting diagram this is

\begin{tikzcd}
[
  column sep={between origins, 60pt},
  row sep={between origins, 40pt}
]
  \{x,y\}
  \ar[dd]
  \ar[rr]
  &&
  {[0,1]}\vee_{\{0,1\}} \{x\leftrightarrow 0, 1\leftrightarrow x\}
  \ar[dd]
  \\
  \\
  X
  \ar[
    rr
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
  \big\{
    x
      \leftrightarrow
    e_{[0,1]}
      \leftrightarrow
    y
  \big\}
\end{tikzcd}

Every completely Hausdorff space is also $ T2\frac{1}{2} $
because there is a surjection
${[0,1]}\vee_{\{0,1\}} \{x\leftrightarrow 0, 1\leftrightarrow y\}
  \to
 \{\boxed{\overset{\boxed{x}}{}{\searrow}\underset{U}{}{\swarrow}\overset{\boxed{X}}{}
}\!\!\!\!\!\!\!
\boxed{
{\,\,\,\,\,\,\searrow}\underset{V}{}{\swarrow}\overset{\boxed{y}}{}
}
\}
$.

### Regular spaces

$X$ is [[regular topological space|regular]] if, given any point ${x}$ and closed subset $F$ in $X$ such that ${x}$ does not belong to $F$, they are separated by neighbourhoods, i.e.
   $$
      \{x\} \longrightarrow  {X}
      \;\;\;\,⧄\,\;\;\;
\{
\boxed{
 \boxed{
\overset{\boxed{x}}{}
\searrow
\underset{X}{} \swarrow
\overset{\boxed{U}}{}
}\!\!\!\!\!\!\!
{\,\,\,\,\,\,\searrow\underset{F}{} }
}
\}
        \longrightarrow
      \{\overset{\boxed{x=X=U}}{}\searrow\underset{F}{}\}
   $$
As a lifting diagram
\begin{tikzcd}
[
  column sep={between origins, 60pt},
  row sep={between origins, 40pt}
]
  \{x\}
  \ar[dd]
  \ar[rr]
  &&
\{
\boxed{
 \boxed{
\overset{\boxed{x}}{}
\searrow
\underset{X}{} \swarrow
\overset{\boxed{U}}{}
}\!\!\!\!\!\!\!
{\,\,\,\,\,\,\searrow\underset{F}{} }
}
\}
  \ar[dd]
  \\
  \\
  X
  \ar[
    rr
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&
      \{\overset{\boxed{x=X=U}}{}\searrow\underset{F}{}\}
\end{tikzcd}

Indeed, in the interesting  case that $x$ maps to $x$
by the top horizontal arrow,
the separating neighbourhoods would be
the preimage under the diagonal arrow of
$\{\boxed{x}\}$ and $\{\overset{\boxed{U}}{}\searrow\underset{F}{}\}$.

   In fact, in a regular space, any such ${x}$ and ${F}$ will also be separated by closed neighbourhoods obtained by "applying" the lifting property again to $\{\overset{\boxed{x}}{}\searrow\underset{X=U=F}{}\}$. Every
regular space is also R1.

### Regular Hausdorff space ($T_3$)

$X$ is a [[regular Hausdorff space]], or T3, if it is both T0 and regular.[1] Every
regular Hausdorff space is also $T2\frac{1}{2}$.

### Completely regular spaces

$X$ is completely regular if, given any point ${x}$ and closed set $F$ in $X$ such that ${x}$ does not belong to $F$, they are separated by a continuous function, i.e.

   $$
     \{x\} \longrightarrow  {X}
     \;\;\;\,⧄\,\;\;\;
     [0,1] \vee_{\{1\}} {\{1\leftrightarrow F\}}
        \longrightarrow  \{ \overset{e_{[0,1]}}{}  \searrow F\}
   $$
Here in $[0,1] \vee_{\{1\}} {\{1\leftrightarrow F\}}$
the points $F$ and $1$ are topologically indistinguishable, $[0,1]$ goes to $x$, and $F$ goes to $F$.

   Every completely regular space is also regular.

### Tychonoff spaces ($T_{3 \tfrac{1}{2}}$)

$X$ is [[Tychonoff space|Tychonoff]], or T3$\frac{1}{2}$, completely T3, or completely regular Hausdorff, if it is both T0 and completely regular.[2] Every Tychonoff space is both regular Hausdorff and completely Hausdorff.

### Normal spaces

$X$ is [[normal topological space|normal]] if any two disjoint closed subsets of $X$ are separated by neighbourhoods, i.e.
   $$
      \emptyset \longrightarrow {X}
      \;\;\;\,⧄\,\;\;\;
      \{\underset{x}{}{\swarrow}
        \overset{U}{}{\searrow}
      \underset{X}{}{\swarrow}\overset{V}{}{\searrow}\underset{y}{}\}
         \longrightarrow
      \{\underset{x}{}{\swarrow}\overset{\boxed{U=X=V}}{}{\searrow}
       \underset{y}{}\}
   $$
As a lifting diagram
\begin{tikzcd}
[
  column sep={between origins, 60pt},
  row sep={between origins, 40pt}
]
  \varnothing
  \ar[dd]
  \ar[rr]
  &&
\{\underset{x}{}{\swarrow}
        \overset{U}{}{\searrow}
      \underset{X}{}{\swarrow}\overset{V}{}{\searrow}\underset{y}{}\}
  \ar[dd]
  \\
  \\
  X
  \ar[
    rr
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&    \{\underset{x}{}{\swarrow}\overset{\boxed{U=X=V}}{}{\searrow}
       \underset{y}{}\}
  \end{tikzcd}

Indeed, the disjoint closed subsets are the preimges of the closed points
$x$ and $y$ under the bottom horizontl arrow, and
their separating neighbourhoods are the preimages of
open subsets $\big\{\underset{x}{}{\swarrow}
        \overset{U}{}\big\}$
and $\big\{\overset{V}{}{\searrow}\underset{y}{}\big\}$.

   In fact, by [[Urysohn's lemma]] a space is normal if and only if any two disjoint closed sets can be separated by a continuous function, i.e.

   $$
      \emptyset \longrightarrow  {X}
      \;\;\;\,⧄\,\;\;\;
       [0,1]\vee_{\{0,1\}} \{0'\leftrightarrow 0, 1\leftrightarrow 1'\}
      \longrightarrow  \{0=0'{\searrow}e_{[0,1]}{\swarrow}1=1'\}
   $$
Here in $[0,1]\vee_{\{0,1\}} \{0'\leftrightarrow 0, 1\leftrightarrow 1'\}$
   the points $0',0$ and $1,1'$ are topologically indistinguishable, $[0,1]$ goes to $e_{[0,1]}$, and both $0,0'$ map to point $0=0'$,  and both $1,1'$ map to point $1=1'$.

### Normal Hausdorff spaces ($T_4$)

$X$ is normal Hausdorff, or T4, if it is both T1 and normal. Every normal
Hausdorff space is both Tychonoff and normal regular.

### Completely normal spaces

$X$ is _completely or heriditarily normal_ if any two separated sets $A$ and $B$ are separated by neighbourhoods $U\supset A$ and $V\supset B$ such that $U$ and $V$ do not intersect, i.e.
   $$
     \emptyset \longrightarrow  {X}
     \;\;\;\,⧄\,\;\;\;
     \{
      \underset{X}{}{\swarrow}
     \overset{A\leftrightarrow U}{}
     {\searrow} \underset{U'}{}{\swarrow}
     \overset{W}{} {\searrow}
     \underset{V'}{}{\swarrow}
     \overset{V\leftrightarrow B}{}{\searrow}
     \underset{X}{} \} \longrightarrow  \{U=U',V'=V\}
   $$

   Every completely normal space is also normal.

It is easy to check that a closed inclusion into a heridarily normal space has the same lifting property. As left lifting property is stable under colimits, this implies that a filtered colimit (transfinite composition)
of heriditarily normal spaces is heriditarily normal.
Compare this argument to [[colimits of normal spaces]].

### Perfectly normal spaces

$X$ is [[perfectly normal topological space|perfectly normal]] if any two disjoint closed sets are precisely separated by a continuous function, i.e.

   $$
    \emptyset\longrightarrow {X}
      \;\;\;\,⧄\,\;\;\;
    [0,1]\longrightarrow \{0{\swarrow}X{\searrow}1\}
   $$

   where $(0,1)$ goes to the open point $X$, and $0$ goes to $0$, and $1$ goes to $1$.

   Every perfectly normal space is also completely normal.

## Applications.

The following notation and terminology helps to discuss applications.

### Quillen negation

For a class $P$ of morphisms in a category, its _left Quillen negation_ $P^{⧄ l}$ with respect to the lifting property, respectively its _right Quillen negation_ $P^{⧄ r}$,
is the class of all morphisms which have the left, respectively right, lifting property with respect to each morphism in the class $P$. In notation,
$$
P^{⧄ l} := \{ i \,\,:\,\, \forall p\in P\,\, i\;\;\;\,⧄\,\;\;\; p\},
P^{⧄r} := \{ p \,\,:\,\, \forall i\in P\,\, i\;\;\;\,⧄\,\;\;\; p\}, P^\lr:=(P^\lrl)^\rlr,..$$

As the terminology might suggest,
taking the Quillen negation of a class/property $P$ is a simple way to define a class of morphisms excluding non-isomorphisms
from $P$, in a way which is useful in a diagram chasing computation, and is often used to
define properties of morphisms starting from an explicitly given class of (counter)examples.
A number of elementary notions may also be expressed using the lifting property starting from a list of (counter)examples.

### Good behaviour under (co)limits

Reformulations in terms of lifting properties clarify behaviour
with respect to (co)limits, as notions defined by lifting properties are closed under (co)limits.

For example, as remarked above, it is easy to check that
a closed inclusion into a heridarily normal space has the same lifting property; this is not true for the lifting property defining normality.
Hence,
filtered colimit (transfinite composition)
of closed inclusions of heriditarily normal spaces is heriditarily normal.
Compare this argument to what is given in [[colimits of normal spaces]].

The same argument proves the
[Tychonoff theorem that a product of compact spaces is compact](Tychonoff+theorem#ProofViaTaimanovTheoremAndLiftingProperties).

### Reflection and weak factorisation systems

Lifting properties give rise to reflection and weak factorisation systems.
For example, each map $X\to\{o\}$ decomposes as
$$X \xrightarrow{(T_i)^{⧄rl}} X'\xrightarrow{(T_i)^{⧄l}}\{o\}$$
where $(T_i)$ denotes the morphism appearing in the definition of axiom $T_i$, for $i0,1$. This gives a statement about reflection
close to [[separation axioms#Reflection]].

In terms of Quillen negation we can understand the [[compactly generated topological space#kSpacesAreTheColimitsInTopOfCompactHausdorffSpaces|$k$-coreflection $kTop\to Top$ from the category of compactly generated spaces]] as follows.

Let $(CHaus)=\big(\{0,1\}\to\{0=1\}\cup\{\varnothing\to K\,\,:\,\,K\,\,\text{is compact Hausdorff}\,\}\big)$.

First notice it holds
$k(X) \xrightarrow{(CHaus)^{⧄r}}X$.

Now consider a decomposition of a map $\varnothing\to X$  as
$$\varnothing \xrightarrow{(CHaus)^{⧄rl}} X_{c.g.}\xrightarrow{(CHaus)^{⧄r}}X$$
such that the latter arrow is injective.

It is easy to see that
$X_{c.g.}\xrightarrow{(CHaus)^{⧄r}}X$
means that a subspace of $X_{c.g}$
is closed
whenever it is closed in $X$ and
its intersection with every compact Hausdorff subsets of (the original topology on) $X$ is closed (in the original topology on $X$).
Then $X_{c.g.}$ has all the same closed sets and possibly more,
hence all the same open sets and possibly more.

On the other hand,
$$\varnothing \xrightarrow{(CHaus)^{⧄rl}} X_{c.g.}
\,\,\,⧄\,\,\,\,
k(X)
\xrightarrow{(CHaus)^{⧄r}}X$$
hence the obvious map $X_{c.g.}\to k(X)$ is continuous.

Hence, $X_{c.g.}$ is isomorphic to $k(X)$.

Compare our proof of the following lemmas with the one given in [[subspace topology#pushout]].

+-- {: .num_lemma #pushout}
###### Lemma
The [[pushout]] in [[Top]] of any (closed/open) subspace $i \colon A \hookrightarrow B$ along any [[continuous function]] $f \colon A \to C$,
is a (closed/open) subspace $j: C \hookrightarrow D$.
=--
+-- {: .num_lemma #transfinite}
###### Lemma
Let $\kappa$ be an [[ordinal]], viewed as a [[preorder]] category, and let $F: \kappa \to Top$ be a functor that preserves [[directed colimits]]. Then if $F(i \leq j)$ is a (closed/open) subspace inclusion for each morphism $i \leq j$ of $\kappa$, then the canonical map $F(0) \to colim_{i \in \kappa} F(i)$ is also a (closed/open) inclusion.
=--
\begin{proof} This follows from the fact that being a (closed/open) inclusion is a left lifting property:

*         $(\emptyset\longrightarrow \{o\})^{rr}=\{\{x\leftrightarrow y\rightarrow c\}\longrightarrow\{x=y=c\}\}^l=\{\{x\leftrightarrow y\leftarrow c\}\longrightarrow\{x=y=c\}\}^l$ is the class of subsets, i.e. injective maps $A\hookrightarrow B$ where the topology on $A$ is induced from $B$

*  $\{ \{z\leftrightarrow  x\leftrightarrow  y\rightarrow c\}\longrightarrow\{z=x\leftrightarrow  y=c\} \}^\lrl  = \{\{c\}\longrightarrow \{o\rightarrow c\}\}^\lr$ is the class of closed inclusions $A\subset B$ where $A$ is closed

*  $\{ \{z\leftrightarrow  x\leftrightarrow  y\leftarrow c\}\longrightarrow\{z=x\leftrightarrow  y=c\} \}^\lrl$ is the class of open inclusions
$A\subset B$ where $A$ is open

\end{proof}

### Urysohn lemma and Tietze extension theorem.

We now explain how to view the standard proof of the [[Urysohn lemma]] in terms of lifting properties.

Let $\Lambda_n=\big\{
\underset{c_0}{} \swarrow
\overset{o_1}{} \searrow \cdots \swarrow
\overset{o_n}{} \searrow
\underset{c_n}{} \big\} $ for $n\geq 1$. Pick a sequence of  "subdivision" maps, e.g.
$$\Lambda_{2({n+1})}\to \Lambda_{2n}$$
$$c_{2i}\mapsto c_i,\,\,\, o_{2i+1},c_{2i+1},o_{2i+2}\mapsto o_i,\,\,\, 0\leq i\leq n-1,\,\,\,c_{2n}\mapsto c_n,$$
and let $\Lambda_\infty=\lim_{n\to\infty} \Lambda_{2^n},$ be the limit in the category of topological spaces.

In this notation normality (T4) is defined by $\varnothing \to X \;\;\;\,⧄\,\;\;\; \Lambda_2\to \Lambda_1$.
Iterating the lifting property shows that $f \;\;\;\,⧄\,\;\;\; \Lambda_2\to \Lambda_1$ implies
that $f \;\;\;\,⧄\,\;\;\; \Lambda_{2n}\to \Lambda_n$ and, passing to the limit,
$f \;\;\;\,⧄\,\;\;\; \Lambda_\infty\to \Lambda_1$.

Iterating the lifting property $T_4$ implies that
$$\Lambda_\infty\to  \Lambda_1 \in \{ \Lambda_2 \to \Lambda_1\}^{lr}.$$

The standard arguments from the proof of Urysohn lemma give the following
relation between $\Lambda_\infty$ and $\mathbf{R}$.

1. $[0,1]$  and  ${[0,1]}\vee_{\{0,1\}} \{e_F\leftrightarrow 0, 1\leftrightarrow e_G\}$ are retracts of $\Lambda_\infty$.

2. The map $\Lambda_\infty\to \Lambda_1$ factors as
$$\Lambda_\infty\to   {[0,1]}\vee_{\{0,1\}} \{e_F\leftrightarrow 0, 1\leftrightarrow e_G\}\to \Lambda_2 \to \Lambda_1$$

Item (1) implies that
$A \to X \;\;\;\,⧄\,\;\;\; \Lambda_\infty\to \{o\}$ implies $A \to X \;\;\;\,⧄\,\;\;\; [0,1]\to \{o\}$
(and  $A \to X \;\;\;\,⧄\,\;\;\; [0,1]\vee_{\{0,1\}} \{e_F\leftrightarrow 0, 1\leftrightarrow e_G\}\to \{o\}$).

This reminds one of the [[Tietze extension theorem]] but is different: [it is not true that any closed inclusion
in a normal space lifts wrt to $\Lambda_2 \to \Lambda_1$](https://mathoverflow.net/questions/394187/extending-disjoint-open-subsets-of-a-normal-hausdorff-space/394193), although it is true for closed inclusions
into a heriditarily normal space.

\begin{tikzcd}
[
  column sep={between origins, 120pt},
  row sep={between origins, 40pt}
]
  A
  \ar[dd]
  \ar[rr]
  &[-60pt]
  & \Lambda_\infty \ar[rdd]\ar[dd]
  & { {[0,1]}\vee_{\{0,1\}} \{e_F\leftrightarrow 0, 1\leftrightarrow e_G\} }\ar[l]\ar[dd] \ar[r]
  &  \Lambda_2
  \ar[
    ddll,
    crossing over
  ]
  \\
  \\
  B
  \ar[
    rr
  ]
  \ar[
    uurr,
    dashed,
    "{ \exists }"
  ]
  &&\Lambda_1
  & { {[0,1]}\vee_{\{0,1\}} \{e_F\leftrightarrow 0, 1\leftrightarrow e_G\} }\ar[l]
\end{tikzcd}

Item (2) (see the diagram above) shows that  $A\to B \;\;\;\,⧄\,\;\;\; \Lambda_\infty\to \Lambda_1$ implies that
$A\to B \;\;\;\,⧄\,\;\;\; {[0,1]}\vee_{\{0,1\}} \{e_F\leftrightarrow 0, 1\leftrightarrow e_G\}\to \Lambda_1$.
Finally, for $A=\varnothing$ the latter implies that $\varnothing \to B \;\;\;\,⧄\,\;\;\;   \Lambda_2 \to \Lambda_1$.
This implies the Urysohn lemma that
$$\varnothing \to B \;\;\;\,⧄\,\;\;\;   \Lambda_2 \to \Lambda_1\text{ iff }\varnothing \to B \;\;\;\,⧄\,\;\;\;   {[0,1]}\vee_{\{0,1\}} \{e_F\leftrightarrow 0, 1\leftrightarrow e_G\} \to \Lambda_1$$

The following theorem is a summary of considerations above.
\begin{theorem}

1. $ \{\Lambda_2\to \Lambda_1\} ^{⧄ l} \subset  \{\Lambda_\infty\to \Lambda_1\}^{⧄ l} \subset   \{{[0,1]}\vee_{\{0,1\}} \{e_F\leftrightarrow 0, 1\leftrightarrow e_G\} \to \Lambda_1\}^{⧄ l}  $

2. $ \{\Lambda_2\to \Lambda_1\} ^{⧄ l}  \subset  \{\Lambda_\infty\to \{o\}\}^{⧄ l} \subset   \{{[0,1]} \to \{o\}\}^{⧄ l} $

3. for arbitrary space $B$ it holds
$$\varnothing \to B \;\;\;\,⧄\,\;\;\;   \Lambda_2 \to \Lambda_1\,\,\,\text{ iff }\,\,\,\varnothing \to B \;\;\;\,⧄\,\;\;\;   {[0,1]}\vee_{\{0,1\}} \{e_F\leftrightarrow 0, 1\leftrightarrow e_G\} \to \Lambda_1$$

\end{theorem}

A noted above, item (3) is the usual statement of Urysohn lemma, and item (2) is similar but not equivalent to the Tietze extension theorem.

### Extremally disconnected spaces being projective
 {#ExtremallyDisconnectedSpacesBeingProjective}

The morphism {#DiscussionOfLiftingForExtremallyDisconnected}
$f_{e.d.}:= \boxed{
\boxed{{}^{\boxed{u}}\!\! \searrow_{\,c_1}} \,\,\, \boxed{{}_{c_2}\swarrow^{\boxed{v}}}
}
\to
  \boxed{
    \overset{
      \boxed{
        \boxed{u}
        \;\;
        \,
        \;\;
        \boxed{v}}
      }{
        \underset{
           c
        }
        {
          \searrow \;\, \swarrow
        }
      }
  }
$
is surjective, proper, and has the right lifting property $T_1$.

Both being surjective and [being proper](Tychonoff+theorem#ProofViaTaimanovTheoremAndLiftingProperties) are the right lifting properties.
This means that each morphism  $S\xrightarrow { (f_{e.d.})^{lr} } X$ is surjective and proper,
and if $X$ has $T_1$, so does $S$. In particular, if $X$ is compact Hausdorff, so is $S$.

Each morphism $\varnothing \to X$ can be decomposed as
$$
\varnothing \xrightarrow{ (f_{e.d.})^l } S \xrightarrow { (f_{e.d.})^{lr} } X
$$

This means there is a surjective proper map onto each space from an extremally disconnected space,
and that in this both spaces can be assumed compact Hausdorff.

Gleasons theorem ([here](extremally+disconnected+topological+space#GleasonTheorem)) which says that extremally disconnected spaces are [[projective objects|projective]] in the category
of topological spaces and [[proper maps]], can be expressed by saying that
$\varnothing \to  S \in (proper)^{l}$ whenever $\varnothing \xrightarrow{ (f_{e.d.})^l } S$.

In fact there is a proper morphism $f_{proper}$ of finite spaces such that
for compact Hausdorff spaces it holds
$S \xrightarrow { (f_{proper})^{lr} } X$, i.e. $(f_{proper})^{lr} $
is a class of proper maps containing (necessarily proper) maps of compact Hausdorff spaces.

## Related concepts

* [Tychonoff theorem -- Proof via lifting properties](Tychonoff+theorem#ProofViaTaimanovTheoremAndLiftingProperties)

* [[colimits of normal spaces]]

## References

* {#Gavrolovich14} [[Misha Gavrilovich]], *Point set topology as diagram chasing computations*,  The De Morgan Gazette. 2014. Vol. 5. No. 4. P. 23-32.  ([arXiv:1408.6710](https://arxiv.org/abs/1408.6710), [pdf](http://mishap.sdf.org/mints-lifting-property-as-negation-DMG_5_no_4_2014.pdf))

* [[Misha Gavrilovich]], *The unreasonable power of the lifting property in elementary mathematics*, 2017 ([arXiv:1707.06615](https://arxiv.org/abs/1707.06615),  [pdf](http://mishap.sdf.org/by:gavrilovich/expressive_power_of_the_lifting_property.pdf))

* [[Misha Gavrilovich]], Konstantin Pimenov. *A naive diagram chasing approach to formalisation of tame topology.*, 2018 ([pdf](http://mishap.sdf.org/mintsGE.pdf))

* [[Misha Gavrilovich]], *Extremally disconnected spaces as $\{\{u\to a,b\leftarrow v\}\to\{u\to a=b\leftarrow v\}\}^{lr}$, and being proper as $(\{\{o\}\to\{o\to c\}\}^r_{\le 4})^{lr}$*, 2021 ([pdf](http://mishap.sdf.org/yetanothernotanobfuscatedsyntax.pdf))


separation logic

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### Context
#### Computing
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## Idea

**Separation logic** is an extension of [[Hoare logic]] used to reason about memory separation and memory sharing in [[programs]] with pointer structures.

## References

* Wikipedia, *[Separation logic](https://en.wikipedia.org/wiki/Separation_logic)*

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