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 $\operatorname{St}(A,\mathcal U):=\bigcup\{U\in\mathcal U:U\cap A\ne\varnothing\}.$ 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 $\operatorname{St}(V,\mathcal V)\subseteq U$.

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.