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 strongly paracompact if every open cover of $X$ has a star-finite open refinement. A space $X$ is called fully normal if every open cover of $X$ has a barycentric open refinement. By a theorem of A.H. Stone, for a T₁ space being fully normal and being paracompact are equivalent. There are paracompact $T_1$ spaces which are not strongly paracompact.

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 ${𝒱}.$

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.