Urysohn ($T_{2\frac{1}{2}}$) space: distinct points have neighbourhoods with disjoint closures

Let $(X,\mathcal{T})$ be a topological space (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison), with closures as in Interior, closure, boundary, exterior, derived set and isolated point in a topological space. Then $X$ is an Urysohn space, also written $T_{2\frac{1}{2}}$, when any two distinct points have open neighbourhoods whose closures are disjoint: for all $x,y\in X$ with $x\mathrm{\ne }y$ there are $U,V\in \mathcal{T}$ with

$x\in U,y\in V,\bar{U}\cap \bar{V}=\mathrm{\varnothing }\mathrm{.}$

Equivalently, by Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open, distinct points have disjoint closed neighbourhoods: if $U$ and $V$ are as displayed then $\bar{U}$ and $\bar{V}$ are disjoint closed neighbourhoods of $x$ and $y$; conversely disjoint closed neighbourhoods $K\ni x$ and $L\ni y$ contain open $U\ni x$ and $V\ni y$ with $\bar{U}\subseteq K$ and $\bar{V}\subseteq L$, since $K$ and $L$ are closed, so the closures are disjoint.

The condition is vacuous for a space with at most one point, and nothing is asserted about equal points.

The condition strictly strengthens the Hausdorff condition on its face: $U\subseteq \bar{U}$ and $V\subseteq \bar{V}$, so disjointness of the closures forces disjointness of $U$ and $V$ and hence the Hausdorff property (Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not). That implication is proved as the next item, together with the implication that puts Urysohn spaces below the regular $T_1$ spaces. This page does not exhibit a Hausdorff space that is not Urysohn, and it does not assert that one exists: every witness reachable from the material developed here would need machinery this page does not have, so the question whether the implication reverses is left open here.

A live naming collision, flagged here and settled in this page's conventions. Two different conditions travel under Urysohn's name:

• the one defined above, separation of points by disjoint closed neighbourhoods, which is what this library calls Urysohn and $T_{2\frac{1}{2}}$;
• separation of points by a continuous real-valued function, which is usually called completely Hausdorff and which this library does not define.

Some texts exchange the two names. Neither is Urysohn's lemma, a theorem about normal spaces that is not proved on this page at all (Conventions on this page, and the one implication of the classical chain that is not available at this point in the reading order).

• The fractional numeral is an interpolation, not an arithmetic fact. It records that the condition sits between $T_2$ and $T_3$ in the standard ordering and carries no other meaning; the same is true of $T_{3\frac{1}{2}}$ later on this page.

• Closures, not interiors. Replacing "$\bar{U}\cap \bar{V}=\mathrm{\varnothing }$" by "$U\cap V=\mathrm{\varnothing }$" gives back the Hausdorff condition exactly, so the whole content of the axiom is the passage to closures.