Completely normal ($T_5$) and perfectly normal ($T_6$) spaces

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).

• $X$ is completely normal when any two separated sets can be put into disjoint open sets: for all $A, B \subseteq X$ that are separated (Separated sets: $\overline{A} \cap B = A \cap \overline{B} = \varnothing$) there are $U, V \in \mathcal{T}$ with $A \subseteq U, \qquad B \subseteq V, \qquad U \cap V = \varnothing .$ $X$ is $T_5$ when it is completely normal and $T_1$ ($T_0$ (Kolmogorov) and $T_1$ (Frechet) spaces).
• $X$ is perfectly normal when $X$ is normal (Normal spaces and $T_4$ spaces, with the source disagreement over whether normality includes $T_1$ stated explicitly) and every closed subset of $X$ is a $G_\delta$ ($G_\delta$ and $F_\sigma$ subsets of a topological space, agreeing with the real-line notion). $X$ is $T_6$ when it is perfectly normal and $T_1$.

As with regular and normal, neither adjective carries a $T_1$ hypothesis in this library, and the numerals name the conjunctions.

The $G_\delta$ condition, restated by complementation. Every closed subset of $X$ is a $G_\delta$ if and only if every open subset of $X$ is an $F_\sigma$, because complementation exchanges the two classes and exchanges open with closed ($G_\delta$ and $F_\sigma$ subsets of a topological space, agreeing with the real-line notion). Both forms are used below, and the second is the one the implication $T_6 \Rightarrow T_5$ consumes.

Complete normality really is stronger than normality, on its face. Disjoint closed sets are separated (Separated sets: $\overline{A} \cap B = A \cap \overline{B} = \varnothing$), so the complete-normality condition applies in particular to them; that is the whole proof of the next item. What complete normality adds is the ability to separate sets that are not closed, for instance the two sets $(0,1)$ and $(1,2)$ of $\mathbb{R}$, which are separated and neither of which is closed.

A competing definition of perfectly normal, and why this library does not use it. Some texts define a perfectly normal space to be a normal space in which every closed set is a zero set (Zero sets and cozero sets of continuous real-valued functions). That condition is equivalent to the one above, but the equivalence rests on Urysohn's lemma, which is not available at this point in the reading order; the $G_\delta$ form is therefore the definition here, and no statement on this page asserts the equivalence. What is proved here is one direction in the metric case, where the distance function exhibits every closed set simultaneously as a zero set and as a $G_\delta$.

• Both axioms are about pairs of sets, not about points. Neither implies $T_0$: the indiscrete topology on a two-point set is completely normal and perfectly normal, since its only separated pairs have an empty member and its only closed sets are open, and it is not $T_0$. That is why the numerals $T_5$ and $T_6$ include $T_1$.

• A frequently quoted equivalent of complete normality is not proved here. A space is completely normal exactly when every subspace of it is normal, which is why hereditarily normal is the other common name. This page defines and uses only the separated-sets form; the hereditary characterisation belongs to a later page, and nothing here depends on it.

• The chain at the top. Perfectly normal implies completely normal, which implies normal; the second implication is immediate and the first is a real theorem, proved two items below.