Star refinement

The general definition makes sense for arbitrary coverings and does not require a topology. Let ${\displaystyle X}$ be a set and let ${\displaystyle {\mathcal {U}}}$ be a covering of ${\displaystyle X,}$ that is, ${\textstyle X=\bigcup {\mathcal {U}}.}$ Given a subset ${\displaystyle S}$ of ${\displaystyle X,}$ the star of ${\displaystyle S}$ with respect to ${\displaystyle {\mathcal {U}}}$ is the union of all the sets ${\displaystyle U\in {\mathcal {U}}}$ that intersect ${\displaystyle S,}$ that is, $${\displaystyle \operatorname {st} (S,{\mathcal {U}})=\bigcup {\big \{}U\in {\mathcal {U}}:S\cap U\neq \varnothing {\big \}}.}$$

Given a point ${\displaystyle x\in X,}$ we write ${\displaystyle \operatorname {st} (x,{\mathcal {U}})}$ instead of ${\displaystyle \operatorname {st} (\{x\},{\mathcal {U}}).}$

A covering ${\displaystyle {\mathcal {U}}}$ of ${\displaystyle X}$ is a refinement of a covering ${\displaystyle {\mathcal {V}}}$ of ${\displaystyle X}$ if every ${\displaystyle U\in {\mathcal {U}}}$ is contained in some ${\displaystyle V\in {\mathcal {V}}.}$ The following are two special kinds of refinement. The covering ${\displaystyle {\mathcal {U}}}$ is called a barycentric refinement of ${\displaystyle {\mathcal {V}}}$ if for every ${\displaystyle x\in X}$ the star ${\displaystyle \operatorname {st} (x,{\mathcal {U}})}$ is contained in some ${\displaystyle V\in {\mathcal {V}}.}$^[1][2] The covering ${\displaystyle {\mathcal {U}}}$ is called a star refinement of ${\displaystyle {\mathcal {V}}}$ if for every ${\displaystyle U\in {\mathcal {U}}}$ the star ${\displaystyle \operatorname {st} (U,{\mathcal {U}})}$ is contained in some ${\displaystyle V\in {\mathcal {V}}.}$^[3][2]

There is another, related but distinct concept. An open cover is star-finite if each member of ${\displaystyle U\in {\mathcal {U}}}$ meets only finitely many members of ${\displaystyle {\mathcal {U}}}$. A space ${\displaystyle X}$ is called strongly paracompact if every open cover of ${\displaystyle X}$ has a star-finite open refinement. A space ${\displaystyle X}$ is called fully normal if every open cover of ${\displaystyle X}$ has a barycentric open refinement.

Every star refinement of a cover is a barycentric refinement of that cover. The converse is not true, but a barycentric refinement of a barycentric refinement is a star refinement.^[4][5][6][7]

Given a metric space ${\displaystyle X,}$ let ${\displaystyle {\mathcal {V}}=\{B_{\epsilon }(x):x\in X\}}$ be the collection of all open balls ${\displaystyle B_{\epsilon }(x)}$ of a fixed radius ${\displaystyle \epsilon >0.}$ The collection ${\displaystyle {\mathcal {U}}=\{B_{\epsilon /2}(x):x\in X\}}$ is a barycentric refinement of ${\displaystyle {\mathcal {V}},}$ and the collection ${\displaystyle {\mathcal {W}}=\{B_{\epsilon /3}(x):x\in X\}}$ is a star refinement of ${\displaystyle {\mathcal {V}}.}$

By a theorem of A.H. Stone, for a T₁ space being fully normal and being paracompact are equivalent.

There are paracompact ${\displaystyle T_{1}}$ spaces which are not strongly paracompact.^[8] Every locally separable metric space is strongly paracompact. Conversely every connected, strongly paracompact metric space is separable, so non-separable Banach spaces are a typical example of metric spaces which fail to be strongly paracompact.

• Family of sets – Any collection of sets, or subsets of a set