Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not

A topological space $(X, \mathcal{T})$ (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison) is Hausdorff when any two distinct points are separated by disjoint open sets: for all $x, y \in X$ with $x \ne y$ there are $U, V \in \mathcal{T}$ with

$x \in U, \qquad y \in V, \qquad U \cap V = \varnothing .$

Since an open set containing a point is an open neighbourhood of it (Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open), the condition reads: distinct points have disjoint open neighbourhoods. Nothing is asserted about points that are equal, and the condition is vacuous for a space with at most one point, so every such space is Hausdorff.

Every metrizable space is Hausdorff. This is not proved here, because it is already discharged: Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not records it among the two things every metrizable space has, deriving it from Distinct points of a metric space have disjoint balls around them, which separates $p \ne q$ in a metric space by the disjoint open balls $B(p,r)$ and $B(q,r)$ with $r = d(p,q)/2 > 0$. In particular $\mathbb{R}$ with its usual topology, every $\mathbb{R}^n$, and every subspace of a metrizable space are Hausdorff.

Not every space is Hausdorff. The indiscrete topology $\mathcal{T}_{\mathrm{ind}} = \{\varnothing, X\}$ on a set $X = \{a,b\}$ with $a \ne b$ (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies) is not: the only open set containing $a$ is $X$, the only one containing $b$ is $X$, and $X \cap X = X \ne \varnothing$. This is the same two-point space that Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not uses to exhibit a topology induced by no metric, and the reason is the same one: failure of the Hausdorff condition is an obstruction to metrizability.

Being Hausdorff is a topological property (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological). If $h : X \to Z$ is a homeomorphism and $X$ is Hausdorff, then for $z \ne z'$ in $Z$ the points $h^{-1}(z)$ and $h^{-1}(z')$ are distinct, so they have disjoint open $U, V$; the images $h[U]$ and $h[V]$ are open, disjoint, and contain $z$ and $z'$ respectively, a homeomorphism carrying the open sets of one space bijectively onto those of the other. So no space homeomorphic to a Hausdorff space fails the condition.

Scope of this item. Only the definition, the metrizable case and the two-point failure are recorded here, because that is all this page uses. The Hausdorff condition is one of a graded family of separation axioms; that family, its ordering, and the questions of which of its members are hereditary or preserved by products, are not available at this point in the reading order and nothing here anticipates them. What this page does use is a single negative result: a quotient of a Hausdorff space need not be Hausdorff, which is recorded below as a false statement and witnessed on the companion page.

• Hausdorff spaces have closed singletons. Fix $x \in X$ and take the union of all open subsets of $X$ that avoid $x$. Every $y \ne x$ belongs to one of them, by Hausdorff separation of $x$ and $y$, while $x$ belongs to none. The union is therefore exactly $X \setminus \{x\}$, so $\{x\}$ is closed. Thus the Hausdorff property implies the singleton-closed ($T_1$) property. The converse fails: closed singletons need not give disjoint neighbourhoods of distinct points.

• What the Hausdorff condition buys, in the one place this page needs it. Separation of distinct points by disjoint open sets is exactly what a quotient map can destroy: identifying points of a Hausdorff space can leave two classes every pair of whose open neighbourhoods meet, and the companion page exhibits such a quotient of a metrizable space. Nothing weaker than an explicit witness settles that, since the condition is a statement about all pairs of open sets.

• The name. Hausdorff's own 1914 axiom system for a topological space included this condition, so "topological space" once meant what is now called a Hausdorff space; this library follows the modern convention in which Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison assumes no separation at all and every separation hypothesis is stated where it is used.