CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
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.
There are many equivalent ways of characterizing a space $S$ as Hausdorff. The traditional definition is this:
Given points $x$ and $y$ of $S$, if $x \neq y$, then there exist open neighbourhoods $U$ of $x$ and $V$ of $y$ in $S$ that are disjoint: such that their intersection $U \cap V$ is the empty set (or explicitly, such that $x' \ne y'$ whenever $x' \in U$ and $y' \in V$).
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 $S \times S$.
Here is a classically equivalent definition that is more suitable for constructive mathematics:
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 $x = y$.
This is the mundane way of saying that $=$ is closed in $S \times S$.
Another way of saying this, which makes sense also for locales, is the following:
The diagonal embedding $S \to S \times S$ 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:
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.
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.
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).
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:
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.
The only reasonable definition for a locale $X$ to be Hausdorff is that its diagonal $X\to X\times X$ is a closed (and hence proper) inclusion. However, if $X$ is a sober space regarded as a locale, this might not coincide with the condition for $X$ to be Hausdorff as a space, since the product $X\times X$ in the category of locales might not coincide with the product in the category of spaces. But it does coincide if $X$ 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 $X$ regarded as a locale. Since it is locally compact, the locale product $X\times X$ coincides with the space product (a theorem that is valid constructively); but nevertheless we have:
In particular, the statement “all discrete locales are Hausdorff (as locales)” is equivalent to excluded middle.
In terms of ‘weakly closed’ and ‘strongly closed’ subsets, as described at closed subset, Definition 2 says that $=$ is weakly closed in $X \times X$. If we instead say that $=$ is strongly closed, then this means that there is a tight inequality $\ne$ (the symmetrization of the exterior? of $=$) relative to which Definition 1 holds. (We use $\ne$ twice in that definition: in the hypothesis that $x \ne y$ and in the conclusion that $x' \ne y'$.) This corresponds to (3) above for a discrete space.
Comments on the relation to topos theory are for instance in