nLab
Hausdorff space

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

A topological space (or more generally, a convergence space) is Hausdorff if convergence is unique. The concept can also be defined for locales (see Definition 3 below) and categorified (see Beyond topological spaces below). A Hausdorff space is often called T 2T_2, since this condition came second in the original list of four separation axioms (there are more now) satisfied by metric spaces.

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

Hausdorff spaces are a kind of nice topological space; they do not form a particularly nice category of spaces themselves, but many such nice categories consist of only Hausdorff spaces. In fact, Felix Hausdorff's original definition of ‘topological space’ actually required the space to be Hausdorff, hence the name. Certainly homotopy theory (up to weak homotopy equivalence) needs only Hausdorff spaces. It is also common in analysis to assume that all spaces encountered are Hausdorff; if necessary, this can be arranged since every space has a Hausdorff quotient (in fact, the Hausdorff spaces form a reflective subcategory of Top), although usually an easier method is available than this sledgehammer.

Definitions

There are many equivalent ways of characterizing a space SS as Hausdorff. The traditional definition is this:

Definition

Given points xx and yy of SS, if xyx \neq y, then there exist open neighbourhoods UU of xx and VV of yy in SS that are disjoint: sutch that their intersection UVU \cap V is the empty set (or explicitly, such that xyx' \ne y' whenever xUx' \in U and yVy' \in V).

That is, any two distinct points can be separated by open neighbourhoods, and it is simply a mundane way of saying that \ne is open in the product topology on S×SS \times S.

Here is a classically equivalent definition that is more suitable for constructive mathematics:

Definition

Given points xx and yy of SS, if every neighbourhood UU of xx in SS meets every neighbourhood VV of yy in SS (which means that UVU \cap V is inhabited), then x=yx = y.

This is the mundane way of saying that == is closed in S×SS \times S.

Another way of saying this, which makes sense also for locales, is the following:

Definition

The diagonal embedding SS×SS \to S \times S is a proper map (or equivalently a closed map, since any closed subspace inclusion is proper).

This way of stating the definition generalizes to topos theory and thus to many other contexts; but it is not always a faithful generalization of the classical notion for topological spaces. See Beyond topological spaces below for more.

Here is an equivalent definition (constructively equivalent to Definition 2) that makes sense for arbitrary convergence spaces:

Definition

Given a net (or equivalently, a proper filter) in SS, if it converges to both xx and yy, then x=yx = y.

That is, convergence in a Hausdorff space is unique.

Examples

Properties

Hausdorff reflection

Proposition

(Hausdorff reflection)

For every topological space XX there exists a Hausdorff topological space HXH X and a continuous function

h X:XHX h_X \;\colon\; X \longrightarrow H X

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

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

are in bijection with continuous function of the form

f˜:HXY \tilde f \;\colon\; H X \longrightarrow Y

and such that the bijection is constituted by

f=f˜h X:Xh XHXf˜Y. f = \tilde f \circ h_X \;\colon\; X \overset{h_X}{\longrightarrow} H X \overset{\tilde f}{\longrightarrow} Y \,.

Here HXH X (or more precisely h Xh_X) is also called the Hausdorff reflection (or Hausdorffication or similar) of XX.

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

(XfY)(HXHfHY) (X \overset{f}{\to} Y) \;\mapsto\; (H X \overset{H f}{\to} H Y)

by setting

Hf:[x][f(x)], H f \;\colon\; [x] \mapsto [f(x)] \,,

where [x][x] denotes the equivalence class under X\sim_X of any xXx \in X.

Finally, the comparison map is compatible with this in that the follows squares commute:

X f Y h X h Y HX Hf HY. \array{ X &\overset{f}{\longrightarrow}& Y \\ {}^{\mathllap{h_X}}\downarrow && \downarrow^{\mathrlap{h_Y}} \\ H X &\underset{H f}{\longrightarrow}& H Y } \,.
Proof

There are various ways to construct h Xh_X, see below prop. 3 and prop. 4

Remark

In the language of category theory the Hausdorff reflection of prop. 2 says that

  1. HH is a functor H:TopTop HausH \;\colon\; Top \longrightarrow Top_{Haus} from the category Top of topological spaces to the full subcategory Top HausιTopTop_{Haus} \overset{\iota}{\hookrightarrow} Top of Hausdorff topological spaces;

  2. h X:XHXh_X \colon X \to H X is a natural transformation from the identity functor on Top to the functor ιH\iota \circ H

  3. Hausdorff topological spaces form a reflective subcategory of all topological spaces in that HH is left adjoint to the inclusion functor ι\iota

    Top HausιHTop. Top_{Haus} \underoverset{\underset{\iota}{\hookrightarrow}}{\overset{H}{\longleftarrow}}{\bot} Top \,.

There are various ways to see the existence and to construct the Hausdorff reflection. The following is maybe the quickest way to see the existence, even though it leaves the actual construction rather implicit.

Proposition

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 Hausdorff topological space YY we have f(x)=f(y)f(x) = f(y).

Then the set of equivalence classes

HXX/ H X \coloneqq X /{\sim}

equipped with the quotient topology is a Hausdorff topological space and the quotient map h X:XX/h_X \;\colon\; X \to X/{\sim} exhibits the Hausdorff reflection of XX, according to prop. 2.

Proof

First observe that every continuous function f:XYf \colon X \longrightarrow Y into a Hausdorff space YY factors uniquely via h Xh_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) \,.

That this is well defined and continuous follows directly from the definitions.

What remains to be seen is that HXH X is indeed a Hausdorff space. Hence assume that [x][y]HX[x] \neq [y] \in H X. By construction of HXH X this means that there exists a Hausdorff 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 Hausdorff, there exist disjoint open neighbourhoods U x,U yτ YU_x, U_y \in \tau_Y. Moreover, by the previous statement there exists a continuous function f˜:HXY\tilde f \colon H X \to Y with f˜([x])=f(x)\tilde f([x]) = f(x) and f˜([y])=f(y)\tilde f([y]) = f(y). Since, by the nature of continuous functions, the pre-images f˜ 1(U x),f˜ 1(U y)HX\tilde f^{-1}( U_x ), \tilde f^{-1}(U_y) \subset H X are still disjoint and open, we have found disjoint neighbourhoods of [x][x] and [y][y]. Hence HXH X is Hausdorff.

Some readers may find the following a more direct way of constructing the Hausdorff reflection:

Proposition

For (Y,τ Y)(Y,\tau_Y) a topological space, write r YY×Yr_Y \subset Y \times Y for the transitive closure of the 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. 3.

(vanMunster 14, section 4)

Monadicity

The topology on a compact Hausdorff space is given precisely by the (existent because compact, unique because Hausdorff) limit of each ultrafilter on the space. Accordingly, compact Hausdorff topological spaces are (perhaps surprisingly) described by a (large) algebraic theory. In fact, the category of compact Hausdorff spaces is monadic (over Set); the monad in question maps each set to the set ultrafilters on it. (The results of this paragraph require the ultrafilter theorem, a weak form of the axiom of choice; see ultrafilter monad.)

A compact Hausdorff locale (or space) is necessarily regular; a regular locale (or T 0T_0 space) is necessarily Hausdorff. Accordingly, locale theory usually speaks of ‘compact regular’ locales instead of ‘compact Hausdorff’ locales, since the definition of regularity is easier and more natural. Then a version of the previous paragraph works for compact regular locales without the ultrafilter theorem, and indeed constructively over any topos.

Sobriety

Using classical logic (but not in constructive logic) every Hausdorff space is a sober topological space: Hausdorff implies sober.

Relation to compact spaces

Arguably, the desire to make spaces Hausdorff (T 2T_2) in analysis is really a desire to make them T 0T_0; nearly every space that arises in analysis is at least regular, and a regular T 0T_0 space must be Hausdorff. Forcing a space to be T 0T_0 is like forcing a category to be skeletal; indeed, forcing a preorder to be a partial order is a special case of both (see specialisation topology for how). It may be nice to assume, when working with a particular space, that it is T 0T_0 but not to assume, when working with a particular underlying set, that every topology on it is T 0T_0.

Whatever one thinks of that, there is a non-T 0T_0 version of Hausdorff space, an R 1R_1 space. (The symbol here comes from being a weak version of a regular space; in general a T iT_i space is precisely both R i1R_{i-1} and T 0T_0). This is also called a preregular space (in HAF) and a reciprocal space (in convergence theory).

Definition (of R 1R_1)

Given points aa and bb, if every neighbourhood of aa meets every neighbourhood of bb, then every neighbourhood of aa is a neighbourhood of bb. Equivalently, if any net (or proper filter) converges to both aa and bb, then every net (or filter) that converges to aa also converges to bb.

There is also a notion of sequentially Hausdorff space:

Definition (of sequentially Hausdorff)

Whenever a sequence converges to both xx and yy, then x=yx = y.

Some forms of predicative mathematics find this concept more useful. Hausdorffness implies sequential Hausdorffness, but the converse is false even for sequential spaces (although it is true for first-countable spaces).

The reader can now easily define a sequentially R 1R_1 space.

Beyond topological spaces

Hausdorff locales

The most obvious definition for a locale XX to be Hausdorff is that its diagonal XX×XX\to X\times X is a closed (and hence proper) inclusion. However, if XX is a sober space regarded as a locale, this might not coincide with the condition for XX to be Hausdorff as a space, since the Cartesian product X×XX\times X in the category Loc of locales might not coincide with the product in the category Top of tpological spaces (the Tychonoff product). But it does coincide if XX is a locally compact locale, so in that case the two notions of Hausdorff are the same.

Separated toposes and schemes

This notion of a Hausdorff locale is a special case of that of Hausdorff topos in topos theory. This also includes notions such as a separated scheme etc. The corresponding relative notion (over an arbitrary base topos) is that of separated geometric morphism. For schemes see separated morphism of schemes.

In constructive mathematics

In constructive mathematics, the Hausdorff notion multifurcates further, due to the variety of possible meanings of closed subspace. If we ask the diagonal to be weakly closed, then in the spatial case, this means that it contains all its limit points, giving Definition 2 above. But if we ask the diagonal to be strongly closed, i.e. the complement of an open set, then in the spatial case this means that there is a tight inequality \ne (the exterior? of ==) relative to which Definition 1 holds. (We use \ne twice in that definition: in the hypothesis that xyx \ne y and in the conclusion that xyx' \ne y'.)

It is natural to call these conditions weakly Hausdorff and strongly Hausdorff, but one should be aware of terminological clashes: in classical mathematics there is a different notion of a weak Hausdorff space, whereas (strong) Hausdorffness for locales has by some authors been called “strongly Hausdorff” only to contrast it with Hausdorffness for spaces.

As a simple example, consider a discrete space XX regarded as a locale. Since it is locally compact, the locale product X×XX\times X coincides with the space product (a theorem that is valid constructively); but nevertheless we have:

  1. The diagonal XX×XX\to X\times X always has an open complement.
  2. Definition 2 above always holds, since {x}\{x\} and {y}\{y\} are neighborhoods of xx and yy, and if they intersect then x=yx=y. That is, XX is spatially weakly Hausdorff.
  3. The diagonal XX×XX\to X\times X is the complement of an open subset (i.e. XX is spatially strongly Hausdorff) if and only if equality in XX is stable under double negation, in other words if XX admits a tight inequality relation.
  4. The locale diagonal Δ:XX×X\Delta:X\to X\times X is a closed sublocale (i.e. XX is localically strongly Hausdorff) if and only if XX has decidable equality. For closedness of Δ\Delta means that Δ *(UΔ *(V))Δ *(U)V\Delta_\ast(U\cup \Delta^\ast(V)) \subseteq \Delta_\ast(U) \cup V for any UO(X)U\in O(X) and VO(X×X)V\in O(X\times X), where xΔ *(V)x\in \Delta^\ast(V) means (x,x)V(x,x)\in V, while (x,y)Δ *(U)(x,y)\in \Delta_\ast(U) means (x=y)(xU)(x=y)\to (x\in U). Now let U=U = \emptyset and V={(x,x)xX}V = \{ (x,x) \mid x\in X \}. Then (x,y)Δ *(UΔ *(V))(x,y) \in \Delta_\ast(U\cup \Delta^\ast(V)) means (x=y)((x=x))(x=y) \to (\bot \vee (x=x)), which is a tautology; while (x,y)Δ *(U)V(x,y) \in \Delta_\ast(U) \cup V means ((x=y))(x=y)((x=y)\to \bot) \vee (x=y), i.e. ¬(x=y)(x=y)\neg(x=y) \vee (x=y).
  5. I don’t know what it means for XX to be localically weakly Hausdorff. (Weak closure in locales is very inexplicit.)

In particular, the statement “all discrete locales are localically strongly Hausdorff” is equivalent to excluded middle.

However, non-discrete spaces can constructively be localically strongly Hausdorff without having decidable equality. For instance, any regular space is also regular as a locale, and hence localically strongly Hausdorff. We can also say:

Theorem

In any topological space XX, let x#yx\#y mean that there exist opens U,VU,V with xUx\in U and yVy\in V and UV=U\cap V = \emptyset; then #\# is always an inequality relation. If the spatial product X×XX\times X coincides with the locale product (such as if XX is locally compact), then XX is localically strongly Hausdorff if and only if #\# is an apartness relation and every open set in XX is #\#-open (i.e. for any xUx\in U and yXy\in X we have yUx#yy\in U \vee x\#y).

Proof

Note that #\# is, as a subset W #X×XW_\# \subseteq X\times X, the exterior of the diagonal in the product topology, mentioned above. If XX is localically strongly Hausdorff, then W #W_\# must be the open set of which the diagonal is the complementary closed sublocale, since it is the largest open set disjoint from the diagonal.

To say that the diagonal is its complementary closed sublocale implies in particular that for any open set UXU\subseteq X, the open set (U×U)W #(U\times U) \cup W_\# is the largest open subset of X×XX\times X whose intersection with the diagonal is contained in UU=UU\cap U = U. Specifically, therefore, (U×U)W #(U\times U) \cup W_\# contains U×XU\times X (since U×XU\times X is an open subset of X×XX\times X whose intersection with the diagonal is UU). That is, if xUx\in U and yXy\in X, then either (x,y)U×U(x,y)\in U\times U (i.e. yUy\in U) or (x,y)W #(x,y)\in W_\# (i.e. x#yx\#y). This shows that UU is #\#-open.

To show that #\# is an apartness, note that for any xx the set {zx#z}\{ z \mid x\# z \} is open, since it is the preimage of W #W_\# under a section of the second projection X×XXX\times X \to X. Thus, it is #\#-open, which is to say that if x#zx\# z then for any yy either x#yx\#y or y#zy\#z, which is the missing comparison axiom for #\# to be an apartness.

Conversely, suppose #\# is an apartness and every open set is #\#-open (i.e. the apartness topology refines the given topology on XX). Let AX×XA\subseteq X\times X be an open set; we must show that AW #A\cup W_\# is the largest open subset of X×XX\times X whose intersection with the diagonal is contained in AΔ XA\cap \Delta_X. In other words, suppose U×VU\times V is a basic open in X×XX\times X and (U×V)Δ X(U\times V)\cap \Delta_X (which is UVU\cap V) is contained in AΔ XA\cap \Delta_X; we must show U×VAW #U\times V\subseteq A\cup W_\#. In terms of elements, we assume that if xUx\in U and xVx\in V then (x,x)A(x,x)\in A, and we must show that if xUx\in U and yVy\in V then (x,y)Ax#y(x,y)\in A \vee x\#y.

Assuming xUx\in U and yVy\in V, since UU and VV are #\#-open we have either yUy\in U or x#yx\# y, and either xVx\in V or x#yx\#y. Since we are done if x#yx\#y, it suffices to assume yUy\in U and xVx\in V. Therefore, by assumption, (x,x)A(x,x)\in A and (y,y)A(y,y)\in A. Since AA is open in the product topology, we have opens U,VU',V' with xUx\in U' and yVy\in V' and U×UAU'\times U'\subseteq A and V×VAV'\times V' \subseteq A. But now #\#-openness of UU' and VV' tells us again that either x#yx\#y (in which case we are done) or yUy\in U' and xVx\in V'. In the latter case, (x,y)U×U(x,y)\in U'\times U' (and also V×VV'\times V'), and hence is also in AA.

Note that the apartness #\# need not be tight, and in particular XX need not be spatially Hausdorff. In particular, if XX might not even be T 0T_0: since localic Hausdorffness is (of course) only a property of the open-set lattice, it only “sees” the sobrification and in particular the T 0T_0 quotient (Kolmogorov quotient). However, this is all that can go wrong: if ¬(x#y)\neg(x\# y), then by #\#-openness every open set containing xx must also contain yy and vice versa, so if XX is T 0T_0 then x=yx=y and #\# is tight.

If the locale product X×XX\times X does not coincide with the spatial product, then the “only if” direction of the above proof still works, if we define W #W_\# to be the open part of the locale product X×XX\times X given by W #={UVUV=}W_\# = \bigvee \{ U\otimes V \mid U\cap V = \emptyset \}. A different proof is to recall that by this theorem, an apartness relation is the same as a (strongly) closed equivalence relation on a discrete locale, and the quotient of such an equivalence relation is the #\#-topology. Thus, if XX is localically strongly Hausdorff, its diagonal is a closed equivalence relation, which yields by pullback a closed equivalence relation on the discrete locale X dX_d on the same set of points. This is the kernel pair of the canonical surjection X dXX_d \to X, and hence its quotient (the #\#-topology) maps to XX, i.e. refines the topology of XX.

References

Due to

See also

A detailed discussion of Hausdorff reflection is in

  • Bart van Munster, The Hausdorff quotient, 2014 (pdf)

Comments on the relation to topos theory are for instance in

  • S. Niefield, A note on the locally Hausdorff property, Cahiers TGDC (1983) (numdam)

Revised on June 30, 2017 05:49:39 by Daniel Luckhardt (134.76.82.216)