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, in the definition of a strongly paracompact 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 $V\in {𝒱}.$ 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 $V\in {𝒱}.$ A space $X$ is called fully normal if every open cover of $X$ has a barycentric open refinement.

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

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.

There are paracompact $T_1$ spaces which are not strongly paracompact. 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.

See also

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

References

• Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485. 
• Willard, Stephen (2004). General Topology. Dover Books on Mathematics (First ed.). Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.