Normalized families and collectionwise normality

Let $(X,\mathcal{T})$ be a topological space and let $\mathcal{A}=\{A_i:i\in I\}$ be a family of subsets of $X$ that is pairwise disjoint: $A_i\cap A_j=\mathrm{\varnothing }$ for all distinct $i,j$.

• $\mathcal{A}$ is separated when it has a pairwise disjoint open expansion: there are open sets $U_i\supseteq A_i$ with $U_i\cap U_j=\mathrm{\varnothing }$ for all distinct $i,j$. For $I=\mathrm{\varnothing }$ this is vacuous.

• $\mathcal{A}$ is normalized when every subunion can be separated from its complementary subunion: for every $J\subseteq I$ there are disjoint open $U,V\subseteq X$ with ${\bigcup }_{i\in J}{A}_i\subseteq U,{\bigcup }_{i\in I\setminus J}{A}_i\subseteq V\mathrm{.}$ The two halves $J$ and $I\setminus J$ play symmetric roles, so it is enough to test one representative of each complementary pair.

• $X$ is collectionwise normal (cwn) when every discrete family of closed subsets of $X$ (Discrete families and $\sigma$-locally-finite and $\sigma$-discrete bases) is separated.

• Discrete closed families are normalized in a normal space. Let $\mathcal{F}$ be a discrete family of closed sets and $J$ a set of indices. A discrete family is locally finite (Every discrete family is locally finite, so every $\sigma$-discrete basis is $\sigma$-locally finite), and a locally finite union of closed sets is closed (Locally finite families remain locally finite after taking closures, closure commutes with their union, and a locally finite union of closed sets is closed); hence ${\bigcup }_{i\in J}{F}_i$ and ${\bigcup }_{i\mathrm{\notin }J}{F}_i$ are disjoint closed sets. If $X$ is normal (Normal spaces and $T_4$ spaces, with the source disagreement over whether normality includes $T_1$ stated explicitly) they can be separated by disjoint open sets, so $\mathcal{F}$ is normalized. Normality is thus the two-set case of the normalization of discrete closed families.

• Collectionwise normality implies normality. For disjoint closed $A,B\subseteq X$ the two-member family $\{A,B\}$ is discrete: a point of $A$ has a neighbourhood missing $B$, one of $B$ has a neighbourhood missing $A$, and a point outside $A\cup B$ has a neighbourhood missing both, since $A$ and $B$ are closed. Separating that discrete family yields disjoint open sets containing $A$ and $B$, so every cwn space is normal.

• Separation implies normalization. If $U_i\supseteq A_i$ are pairwise disjoint open sets and $J\subseteq I$, then $U:={\bigcup }_{i\in J}{U}_i$ and $V:={\bigcup }_{i\mathrm{\notin }J}{U}_i$ are disjoint open sets containing the two subunions. Hence every separated family is normalized, and the two-member cases of the two conditions coincide.

• No choice is hidden. The definitions distinguish between "there exist open sets $U_i$" and "there is a family $i\mapsto U_i$" only in the usual way: a separation of a family is a family of open sets, so it is a single function together with its verification, not an application of choice.