Star refinement

In mathematics, specifically in the study of topology and open covers of a topological space X, a star refinement is a particular kind of refinement of an open cover of X. The term has two similar but distinct usages. A related term sometimes used to differentiate the weaker of these two properties is the notion of a barycentric refinement.

Star refinements are used in the definition of a fully normal space, and in one among several equivalent formulations of a uniform space.

Definitions

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

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

A covering $𝒰$ of $X$ is a refinement of a covering $𝒱$ of $X$ if every $U\in 𝒰$ is contained in some $V\in 𝒱.$ The following are two special kinds of refinement. The covering $𝒰$ is called a barycentric refinement of $𝒱$ if for every $x\in X$ the star $\mathrm{st}(x,𝒰)$ is contained in some $V\in 𝒱.$Template:SfnTemplate:Sfn The covering $𝒰$ is called a star refinement of $𝒱$ if for every $U\in 𝒰$ the star $\mathrm{st}(U,𝒰)$ is contained in some $V\in 𝒱.$Template:SfnTemplate:Sfn

A space $X$ is called fully normal if every open cover of $X$ has a barycentric open refinement.

Properties and Examples

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.Template:SfnTemplate:Sfn

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

By a theorem of A.H. Stone, for a T₁ space being fully normal and being paracompact are equivalent. This was a landmark theorem and provided the first proof that metric spaces are paracompact. The proof is difficult, but simpler proofs of the paracompactness of metric spaces were later provided.

See also

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References

• Template:Dugundji Topology
• Template:Willard General Topology