Moore spaces and developments

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).

Stars. For a family $\mathcal{U}$ of subsets of $X$ and a set $A\subseteq X$ put $\mathrm{St}(A,\mathcal{U}):=\bigcup \{U\in \mathcal{U}:U\cap A\mathrm{\ne }\mathrm{\varnothing }\},$ the star of $A$ with respect to $\mathcal{U}$ (Refinements, locally finite families, point-finite families, and star refinements); write $\mathrm{St}(x,\mathcal{U})$ for $\mathrm{St}(\{x\},\mathcal{U})$. If $\mathcal{U}$ covers $X$ then $x\in \mathrm{St}(x,\mathcal{U})$, and $\mathrm{St}(x,\mathcal{U})$ is exactly the union of the members of $\mathcal{U}$ containing $x$; it is open as soon as the members of $\mathcal{U}$ are open.

Development. A development for $X$ is a sequence $({\mathcal{G}}_n{)}_{n\in \mathbb{N}}$ of open covers of $X$ such that for every $x\in X$ and every open $D$ with $x\in D$ there is $n\in \mathbb{N}$ with $\mathrm{St}(x,{\mathcal{G}}_n)\subseteq D\mathrm{.}$ Equivalently: the family $\{\mathrm{St}(x,{\mathcal{G}}_n):n\in \mathbb{N}\}$ is a neighbourhood base at $x$ (Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open). A development is decreasing when ${\mathcal{G}}_{n+1}$ refines ${\mathcal{G}}_n$ for every $n$ (Refinements, locally finite families, point-finite families, and star refinements).

$X$ is developable when it admits a development, and a Moore space is a regular $T_1$ space (Regular spaces and $T_3$ spaces, with the source disagreement over whether regularity includes $T_1$ stated explicitly) that is developable.

Every development can be made decreasing, and then it still is one. Let $({\mathcal{G}}_n)$ be a development and let ${\mathcal{G}}_n^{\mathrm{\prime }}$ be the family of all intersections $G_0\cap \cdots \cap G_n$ with $G_i\in {\mathcal{G}}_i$ for $0\le i\le n$. Each ${\mathcal{G}}_n^{\mathrm{\prime }}$ is an open cover of $X$; a member of ${\mathcal{G}}_m^{\mathrm{\prime }}$ is contained in a member of ${\mathcal{G}}_i$ whenever $i\le m$, so ${\mathcal{G}}_m^{\mathrm{\prime }}$ refines ${\mathcal{G}}_i$ and $\mathrm{St}(x,{\mathcal{G}}_m^{\mathrm{\prime }})\subseteq \mathrm{St}(x,{\mathcal{G}}_i)(i\le m);$ in particular ${\mathcal{G}}_m^{\mathrm{\prime }}$ refines ${\mathcal{G}}_i^{\mathrm{\prime }}$ for $i\le m$. Given $x\in D$ open, choose $n$ with $\mathrm{St}(x,{\mathcal{G}}_n)\subseteq D$; then $\mathrm{St}(x,{\mathcal{G}}_n^{\mathrm{\prime }})\subseteq D$ by the displayed inclusion. So $({\mathcal{G}}_n^{\mathrm{\prime }})$ is a decreasing development, and a space is developable if and only if it has a decreasing development.

Developments give first countability. If $({\mathcal{G}}_n)$ is a development, then for each $x$ the sets $\mathrm{St}(x,{\mathcal{G}}_n)$ are open neighbourhoods of $x$, and every open set containing $x$ contains one of them; hence $\{\mathrm{St}(x,{\mathcal{G}}_n):n\in \mathbb{N}\}$ is a countable local base at $x$ and $X$ is first countable (First countable space: a countable neighbourhood base at every point). In particular every Moore space is first countable.

• Conventions. Regular and normal name separation conditions alone in this library, with $T_1$ written separately; Moore space is defined as regular $T_1$ plus developable.

• Why a development and not a metric. Both structures define the topology by countably many "approximations"; a development survives in spaces that carry no compatible metric, and the whole point of this page is that a later ZFC theorem Collectionwise normal Moore spaces are metrizable concerns collectionwise normal Moore spaces, including the regular $T_1$ hypotheses. It does not assert metrization of arbitrary developable spaces. That later theorem is separate from the ZF definitions and finite normalization argument here.

• The normalization uses no choice. The family ${\mathcal{G}}_n^{\mathrm{\prime }}$ is described by a formula from the given ${\mathcal{G}}_0,\dots ,{\mathcal{G}}_n$, and per point one selects one member of each of finitely many covers, which is finite choice and hence available in $\mathrm{Z}\mathrm{F}$.