nLab
Hausdorff space

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 $T_2$, 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 $S$ as Hausdorff. The traditional definition is this:

Definition

Given points $x$ and $y$ of $X$, if $x \neq y$, then there exist open neighbourhoods $U$ of $x$ and $V$ of $y$ in $S$ such that their intersection $U \cap V$ 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 $x$ and $y$ of $S$, if every neighbourhood $U$ of $x$ in $S$ meets every neighbourhood $V$ of $y$ in $S$ (which means that $U \cap V$ is inhabited), then $x = y$.

A more conceptual way of saying this, which makes sense also for locales, is the following

Definition

The diagonal embedding $S \to S \times S$ 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 $S$, if it converges to both $x$ and $y$, then $x = y$.

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 $T_0$ 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.

Proposition

a CW-complex is a Hausdorff space.

Proposition

compact subspaces of Hausdorff spaces are closed.

Related notions

Arguably, the desire to make spaces Hausdorff ($T_2$) in analysis is really a desire to make them $T_0$; nearly every space that arises in analysis is at least regular, and a regular $T_0$ space must be Hausdorff. Forcing a space to be $T_0$ 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 $T_0$ but not to assume, when working with a particular underlying set, that every topology on it is $T_0$.

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

Definition (of $R_1$)

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

There is also a notion of sequentially Hausdorff space:

Definition (of sequentially Hausdorff)

Whenever a sequence converges to both $x$ and $y$, then $x = y$.

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 $R_1$ 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.

Related concepts

• locally Hausdorff topological space

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)