Refinements, locally finite families, point-finite families, and star refinements

Let $X$ be a topological space. A family $\mathcal{V}$ of subsets of $X$ is a refinement of a family $\mathcal{U}$ when every $V\in \mathcal{V}$ is contained in some $U\in \mathcal{U}$. It is an open refinement when, additionally, every $V\in \mathcal{V}$ is open. A refinement of a cover need not itself cover $X$; when it does, it is called a refining cover.

A family $\mathcal{A}$ of subsets of $X$ is locally finite when every point $x\in X$ has a neighbourhood meeting only finitely many members of $\mathcal{A}$. It is point-finite when every $x\in X$ belongs to only finitely many members of $\mathcal{A}$. Local finiteness implies point-finiteness: a neighbourhood of $x$ meeting only finitely many members contains $x$, so every member containing $x$ is among those finitely many. The converse is not part of the definition and can fail.

For a family $\mathcal{U}$ and a subset $A\subseteq X$, its star about $A$ is $\mathrm{St}(A,\mathcal{U}):=\bigcup \{U\in \mathcal{U}:U\cap A\mathrm{\ne }\mathrm{\varnothing }\}\mathrm{.}$ A cover $\mathcal{V}$ is a star refinement of a cover $\mathcal{U}$ when for every $V\in \mathcal{V}$ there is $U\in \mathcal{U}$ with $\mathrm{St}(V,\mathcal{V})\subseteq U$.

The word “neighbourhood” has the library convention from Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open: it need not itself be open. Replacing it by an open neighbourhood gives the same local-finiteness condition, because every neighbourhood contains an open one about the same point.