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 ${\mathcal {U}}$ be a covering of $X,$ that is, ${\textstyle X=\bigcup {\mathcal {U}}.}$ Given a subset $S$ of $X,$ the star of $S$ with respect to ${\mathcal {U}}$ is the union of all the sets $U\in {\mathcal {U}}$ that intersect $S,$ that is, $$\operatorname {st} (S,{\mathcal {U}})=\bigcup {\big \{}U\in {\mathcal {U}}:S\cap U\neq \varnothing {\big \}}.$$

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

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

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.

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

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

• 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) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.