nLab
Hausdorff space

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.

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 XX, if xyx \neq y, then there exist open neighbourhoods UU of xx and VV of yy in SS such that their intersection UVU \cap V is the empty set.

That is, any two distinct points can be separated by open neighbourhoods.

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.

A more conceptual 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.

Properties

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.

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

The only reasonable 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 product X×XX\times X in the category of locales might not coincide with the product in the category of spaces. But it does coincide if XX is a locally compact locale, so in that case the two notions of Hausdorff are the same.

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, the Hausdorff notion multifurcates further, due to the variety of possible meanings of closed subspace. 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; 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.
  3. The diagonal XX×XX\to X\times X is the complement of an open subset if and only if equality in XX is ¬¬\neg\neg-stable, such as if XX admits a tight apartness relation.
  4. The locale diagonal Δ:XX×X\Delta:X\to X\times X is a closed sublocale 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).

In particular, the statement “all discrete locales are Hausdorff (as locales)” is equivalent to excluded middle.

References

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 November 30, 2016 16:31:22 by Mike Shulman (76.167.222.204)