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 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.

This way of stating the definition generalizes to topos theory and thus to many other contexts. 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 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.

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 July 17, 2014 22:33:31 by Urs Schreiber (82.136.246.44)