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separation axioms

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The separation axioms

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. 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. For example the weakest axiom (called T 0T_0) demands that if two points are distinct as elements of the underlying set of points, then there exists at least one open subset that contains one but not the other.

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 1T_1) or that such a pair of open subsets around two distinct points may in addition be chosen to be disjoint (T 2T_2). This last condition, T 2T_2, also called the Hausdorff condition is the most common among all separation axioms. Often (but by far not always) this is considered by default.

The main separation axioms are these:

the main separation axioms

numbernamestatementreformulation
T 0T_0Kolmogorovgiven two distinct points, at least one of them has an open neighbourhood not containing the other pointevery irreducible closed subset is the closure of at most one point
T 1T_1given two distinct points, both have an open neighbourhood not containing the other pointall points are closed
T 2T_2Hausdorffgiven two distinct points, they have disjoint open neighbourhoodsthe diagonal is a closed map
T >2T_{\gt 2}T 1T_1 and…all points are closed and…
T 3T_3regular Hausdorff…given a point and a closed subset not containing it, they have disjoint open neighbourhoods…every neighbourhood of a point contains the closure of an open neighbourhood
T 4T_4normal Hausdorff…given two disjoint closed subsets, they have disjoint open neighbourhoods…every neighbourhood of a closed set also contains the closure of an open neighbourhood
… every pair of disjoint closed subsets is separated by an Urysohn function

Originally in Tietze 23 the four separation axioms T 2,T 3,T 4,T 5T_2, T_3, T_4, T_5 were considered (see at History below for more); nowadays one considers various more. Besides the extrapolation of the original sequence from T 0T_0 through T 6T_6 (with T 212T_{2\frac{1}{2}} and T 312T_{3\frac{1}{2}} interpolated), there is a similar sequence of axioms called R 0,R 1,R 2,R 3R_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 0T_0. This and more is spelled out below.

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 is of a different nature, but is implied by T 2T_2 and implies T 0T_0. Notice that via their full embdding into locales, sober topological spaces may be understood without reference to their underlying set pf 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

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 SS.

Separation conditions

Fix two sets (subsets) FF and GG of SS.

  • The sets FF and GG are disjoint if their intersection is empty:

    FG=. F \cap G = \empty .
  • They are topologically disjoint if there exists a neighbourhood of one set that is disjoint from the other set:

    (UF,UG=)(VG,FV=). (\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:

    (UF,UG=)(VG,FV=)UF,VG,UG=FV=. (\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.

  • They are separated by neighbourhoods if they have disjoint neighbourhoods:

    UF,VG,UV=. \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:

    UF,VG,Cl(U)Cl(V)=. \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-valued function on the space that maps FF to 00 and GG to 11:

    f:SR,Ff *({0})Gf *({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 [ϵ,ϵ][-\epsilon, \epsilon] and [1ϵ,1+ϵ][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 FF to 00 and GG to 11:

    f:SR,F=f *({0})G=f *({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 FF and GG will be points (identified with their singleton subsets); in that case, one usually says distinct in place of disjoint.

Often FF or GG 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 FF is a (point/closed) set and GG is a (point/closed) set, if FF and GG 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 T_0
Separated T 1 T_1 R 0 R_0
Separated by neighbourhoods T 2 T_2 R 1 R_1
Separated by closed neighbourhoods T 212 T_{2\frac{1}{2}} R 112 R_{1\frac{1}{2}}
Separated by a function Completely T 2 T_2 Completely R 1 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

Reformulation in terms of topological closures

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

Proposition

(T 0T_0 in terms of topological closures)

A topological space (X,τ)(X,\tau) is T 0T_0 precisely if the function Cl({})Cl(\{-\}) from the underlying set of XX to the set of irreducible closed subsets of XX, is injective:

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

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

Proof

Assume first that XX is T 0T_0. Then we need to show that if x,yXx,y \in X are such that Cl({x})=Cl({y})Cl(\{x\}) = Cl(\{y\}) then x=yx = y. Hence assume that Cl({x})=Cl({y})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 xx is the same as the union of open subsets that do not contain yy:

UXopenUX\{x}(U)=UXopenUX\{y}(U) \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, xyx \neq y, then by T 0T_0 one of the above unions would contain xx or yy, while the other would not, in contradiction to the above equality. Hence we have a proof by contradiction.

Conversely, assume that (Cl{x}=Cl{y})(x=y)\left( Cl\{x\} = Cl\{y\}\right) \Rightarrow \left( x = y\right), and assume that xyx \neq y. Hence by contraposition Cl({x})Cl({y})\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 this would mean that xCl({y})x \in \mathrm{Cl}(\{y\}) and that yCl({x})y \in \mathrm{Cl}(\{x\}). But this would imply that Cl({x})Cl({y})Cl(\{x\}) \subset \mathrm{Cl}(\{y\}) and that Cl({y})Cl({x})\mathrm{Cl}(\{y\}) \subset \mathrm{Cl}(\{x\}), hence that Cl({x})=Cl({y})\mathrm{Cl}(\{x\}) = \mathrm{Cl}(\{y\}). This is a proof by contradiction.

Proposition

(T 1T_1 in terms of topological closures)

A topological space (X,τ)(X,\tau) is T 1T_1 precisely if all its points are closed points.

Proof

Assume first that (X,τ)(X,\tau) is T 1T_1. We need to show that for every point xXx \in X we have Cl({x})={x}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 xx is X\{x}X \backslash \{x\}, hence if every point yxy \neq x is member of at least one open subset not containing xx. This is true by T 1T_1.

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

Proposition

(T 2T_2 in terms of topological closures)

A topological space (X,τ X)(X,\tau_X) is T 2T_2=Hausdorff precisely if the image of the diagonal

X Δ X X×X x AAA (x,x) \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×X,τ X×X)(X \times X, \tau_{X \times X}).

Proof

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

Hence if XX is Hausdorff, then the diagonal Δ X(X)X×X\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) every point (x,y)(x,y) not on the diagonal, hence with xyx \neq y, has an open neighbourhood U x×U yU_x \times U_y still not intersecting the diagonal, hence so that U xU y=U_x \cap U_y = \emptyset. Thus (X,τ)(X,\tau) is Hausdorff.

Remark

The characterization of the Hausdorff separation condition via the closure of the diagonal in prop. 3 is the basis for the definition of separated scheme.

Proposition

(T 3T_3 in terms of topological closures)

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

{x}VCl(V)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. 5 to the case that C={x}C = \{x\} (using that by T 1T_1, which is part of the definition of T 3T_3, the singleton subset is indeed closed by prop. 2).

Proposition

(T 4T_4 in terms of topological closures)

A topological space (X,τ)(X,\tau) is normal, precisely if for all closed subsets CXC \subset X with open neighbourhood UCU \supset C there exists a smaller open neighbourhood VCV \supset C whose topological closure Cl(V)Cl(V) is still contained in UU:

CVCl(V)U. C \subset V \subset Cl(V) \subset U \,.
Proof

In one direction, assume that (X,τ)(X,\tau) is normal, and consider CUC \subset U. It follows that the complement of the open subset UU is closed and disjoint from CC:

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

Therefore by assumption of normality of (X,τ)(X,\tau), there exists open neighbourhoods VCV \supset C and WX\UW \supset X \backslash U with

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

But this means that

VX\W V \subset X \backslash W

and since the complement X\WX \backslash W of the open set WW is closed, it still contains the closure of VV, so that we have

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

In the other direction, assume that for every open neighbourhood UCU \supset C of a closed subset CC there exists a smaller open neighbourhood VV with CVCl(V)UC \subset V \subset Cl(V) \subset U. Consider disjoint closed subsets C 1,C 2XC_1, C_2 \subset X. We need to produce disjoint open neighbourhoods for them.

From their disjointness it follows that X\C 2C 1X \backslash C_2 \supset C_1 is an open neighbourhood. Hence by assumption there is an open neighbourhood VV with

C 1VCl(V)X\C 2. C_1 \subset V \subset Cl(V) \subset X \backslash C_2 \,.

Thus VC 1V \supset C_1 and X\Cl(X)C 2X \backslash Cl(X) \supset C_2 are two disjoint open neighbourhoods, as required.

Relations between the axioms

First of all, notice (prop. 2) that the T 1T_1 condition, saying that distinct points are separated, is equivalent to the condition that every point is closed. Thus, T 1T_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 0T_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 normal and regular must be R 3R_3. And since R 3R_3 never appears in the tables above, you can take this as a definition of R 3R_3.

In general, the names in this diagram are:

  • ‘P’ for ‘perfectly’,
  • ‘C’ for ‘complete’,
  • ‘N’ for ‘normal’,
  • ‘R’ (without a subscript) for ‘regular
  • ‘T’ or ‘R’ with a subscript are written and pronounced that way.

Warning: T iT_i for i3i \geq 3 has been used in different ways in the past, and perhaps by some schools still. Also, all of the R iR_i terms are rare. It is safest to say, for example, ‘normal Hausdorff’ for T 4T_4 and clearer to say, for example, ‘normal regular’ for R 3R_3. If you want to avoid the subscript terms entirely, then you can, by doing the above and the following:

  • T 212T_{2\tfrac{1}{2}} = Urysohn
  • T 2T_2 = Hausdorff,
  • T 1T_1 = accessible,
  • T 0T_0 = Kolmogorov,
  • R 1R_1 = reciprocal,
  • R 0R_0 = symmetric.

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

  • R 2R_2 = regular,
  • R 212R_{2\frac{1}{2}} = completely regular.

It would be easy to invent an N iN_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 completely regular Hausdorff (T 312T_{3\frac{1}{2}}).

Reflection

Proposition

(T nT_n-reflection)

Let n{0,1,2}n \in \{0,1,2\}. Then for every topological space XX there exists a T nT_n-topological space T nXT_n X and a continuous function of the forma

t n(X):XT nX t_n(X) \;\colon\; X \longrightarrow T_n X

which is the “closest approximation from the left” to XX by a T nT_n-topological space, in that for YY any T nT_n-space, then continuous functions of the form

f:XY f \;\colon\; X \longrightarrow Y

are in bijection with continuous function of the form

f˜:T nXY \tilde f \;\colon\; T_n X \longrightarrow Y

and such that the bijection is constituted by

f=f˜t n(X):Xt n(X)T nXf˜Y. f = \tilde f \circ t_n(X) \;\colon\; X \overset{t_n(X)}{\longrightarrow} T_n X \overset{\tilde f}{\longrightarrow} Y \,.

Here Xt n(X)T n(X)X \overset{t_n(X)}{\longrightarrow} T_n(X) is called the T nT_n-reflection of XX.

Moreover, the operation T n()T_n(-) extends to continuous functions f:XYf \colon X \to Y

(XfY)(T nXT nfT nY) (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 ng)(T nf)=T n(gf)AA,AAT n(id X)=id T nX (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:XYf \colon X \to Y then

t n(Y)f=T n(f)t n(X) t_n(Y) \circ f = T_n(f)\circ t_n(X)

hence then follows squares commutes:

X f Y t n(X) t n(Y) T nX T nf T nY. \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. 7.

Remark

(reflective subcategories)

In the language of category theory the T nT_n-reflection of prop. 6 says that

  1. T n()T_n(-) is a functor T n:TopTop T nT_n \;\colon\; Top \longrightarrow Top_{T_n} from the category Top of topological spaces to the full subcategory Top T nιTopTop_{T_n} \overset{\iota}{\hookrightarrow} Top of Hausdorff topological spaces;

  2. t n(X):XT nXt_n(X) \colon X \to T_n X is a natural transformation from the identity functor on Top to the functor ιT n\iota \circ T_n

  3. T nT_n-topological spaces form a reflective subcategory of all topological spaces in that T nT_n is left adjoint to the inclusion functor ι\iota; this situation is denoted as follows:

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

Generally, an adjunction between two functors

L:𝒞𝒟:R L \;\colon\; \mathcal{C} \leftrightarrow \mathcal{D} \;\colon\; R

is for all pairs of objects c𝒞c \in \mathcal{C}, d𝒟d \in \mathcal{D} a bijection between sets of morphisms of the form

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

i.e.

Hom 𝒟(L(c),d)ϕ c,dHom 𝒞(c,R(d)) 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” in that they for all pairs of morphisms f:ccf \colon c' \to c and g:ddg \colon d \to d' then the folowing diagram commutes:

Hom 𝒟(L(c),d) ϕ c,d Hom 𝒞(c,R(d)) g()L(f) R(g)()f Hom 𝒞(L(c),d) ϕ c,d Hom 𝒟(c,R(d)). \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 nT_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.

Proposition

(T nT_n-reflection via surjections into T nT_n-spaces)

Let n{0,1,2}n \in \{0,1,2\}. Let (X,τ)(X,\tau) be a topological space and consider the equivalence relation \sim on the underlying set XX for which xyx \sim y precisely if for every surjective continuous function f:XYf \colon X \to Y into any T nT_n-topological space YY we have f(x)=f(y)f(x) = f(y).

Then the set of equivalence classes

T nXX/ T_n X \coloneqq X /{\sim}

equipped with the quotient topology is a T nT_n-topological space, and the quotient map t n(X):XX/t_n(X) \;\colon\; X \to X/{\sim} exhibits the T nT_n-reflection of XX, according to prop. 6.

Proof

First we observe that every continuous function f:XYf \colon X \longrightarrow Y into a T nT_n-topological space YY factors uniquely via t n(X)t_n(X) through a continuous function f˜\tilde f

f=f˜h X f = \tilde f \circ h_X

where

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

To see this, first factor ff through its image f(X)f(X)

f:Xf(X)Y f \;\colon\; X \longrightarrow f(X) \hookrightarrow Y

equipped with its subspace topology as a subspace of YY. It follows that f(X)f(X) is a T nT_n-topological space if YY is.

It follows by definition of t n(X)t_n(X) that the factorization exists at the level of sets as stated, since if x 1,x 2Xx_1, x_2 \in X have the same equivalence class [x 1]=[x 2][x_1] = [x_2] in T nXT_n X, then by definition they have the same image under all continuous surjective functions to a T nT_n-space, hence in particular under Xf(X)X \to f(X). This means that f˜\tilde f as above is well defined. Moreover, it is clear that this is the unique factorization.

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

(t n(X)) 1(f˜ 1(U))=(f˜t n(X)) 1(U)=f(U) (t_n(X))^{-1}(\tilde f^{-1}(U)) = ( \tilde f \circ t_n(X) )^{-1}(U) = f(U)

is open in XX. But this is the case by the assumption that ff is continuous.

What remains to be seen is that T nXT_n X as constructed is indeed a T nT_n-topological space. Hence assume that [x][y]T nX[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 nXT_n X this means that there exists a T nT_n-topological space YY and a surjective continuous function f:XYf \colon X \longrightarrow Y such that f(x)f(y)Yf(x) \neq f(y) \in Y. Accordingly, since YY is T nT_n, there exist the respective kinds of neighbourhoods around these image points in YY. Moreover, by the previous statement there exists a continuous function f˜:T nXY\tilde f \colon T_n X \to Y with f˜([x])=f(x)\tilde f([x]) = f(x) and f˜([y])=f(y)\tilde f([y]) = f(y). By the nature of continuous functions, the pre-images of these open neighbourhoods in YY are still open in XX and still satisfy the required disjunction properties. Therefore T nXT_n X is a T nT_n-space.

Here are alternative constructions of the reflections:

Proposition

(Kolmogorov quotient)

Let (X,τ)(X,\tau) be a topological space. Consider the relation on the underlying set by which x 1x 1x_1 \sim x_1 precisely if neighther x ix_i has an open neighbourhood not containing the other. This is an equivalence relation. The quotient topological space XX/X \to X/\sim by this equivalence relation exhibits the T 0T_0-reflection of XX according to prop. 6.

Proposition

(Hausdorff reflection)

For (Y,τ Y)(Y,\tau_Y) a topological space, write r YY×Yr_Y \subset Y \times Y for the transitive closure of tthe relation given by the topological closure Cl(Δ Y)Cl(\Delta_Y) of the image of the diagonal Δ Y:YY×Y\Delta_Y \colon Y \hookrightarrow Y \times Y.

r YTrans(Cl(Delta Y)). r_Y \coloneqq Trans(Cl(Delta_Y)) \,.

Now for (X,τ X)(X,\tau_X) a topological space, define by induction for each ordinal number α\alpha an equivalence relation r αr^\alpha on XX as follows, where we write q α:XH α(X)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 X 0Δ Xr^0_X \coloneqq \Delta_X;

For a successor ordinal we set

  • r X α+1{(a,b)X×X|(q α(a),q α(b))r H α(X)}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 α\alpha we set

  • r X αβ<αr X βr_X^\alpha \coloneqq \underset{\beta \lt \alpha}{\cup} r_X^\beta.

Then:

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

  2. for this α\alpha then H α(X)=H(X)H^\alpha(X) = H(X) is the Hausdorff reflection from prop. 7.

(vanMunster 14, section 4)

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:

Beyond the classical theory

The axioms T 1T_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 0T_0 if and only if the specialisation order is a partial order.
  • The space is R 0R_0 if and only if the specialisation order is an equivalence relation.
  • The space is T 1T_1 if and only if the sepcification 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 2T_2 and below, or rather their negations, can be easily phrased in terms of the 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 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 2T_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 FF and a neighbourhood UU of FF, we talk about an open set GG and an open set UU such that GUG \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 2T_2. (Note that every locale is T 0T_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 xx is disjoint from a closed set FF becomes the condition that xx belongs to an open set GG.

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

History

In Tietze 23, part B, starting on page 300. 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 2T_2, the second is T 3T_3, the third is T 4T_4, and the fourth is T 5T_5. So while this paper may be the first to consider a hierarchy of separation axioms, it is not the source of our T iT_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 lists the T 1T_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.

References

An original article is

  • Heinrich Tietze, Beitrage zur allgemeinen Topologie. I. Axiome für verschiedene Fassungen des Umgebungsbegriffs, Mathematische Annalen, vol 88, pages 290-311 (1923) (online scan)

Lecture notes include

See also the entry

(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.)

Revised on May 23, 2017 04:05:24 by Urs Schreiber (46.183.103.17)