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. A related concept is the notion of barycentric refinement.

Star refinements are used in the definition of fully normal space and in one definition of uniform space. It is also useful for stating a characterization of paracompactness.

Definitions

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]}

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. ^{[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 ${\displaystyle {\mathcal {V}}.}$

See also

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

Notes

1. Dugundji 1966, Definition VIII.3.1, p. 167.
2. Willard 2004, Definition 20.1.
3. Dugundji 1966, Definition VIII.3.3, p. 167.
4. Dugundji 1966, Prop. VIII.3.4, p. 167.
5. & Willard 2004, Problem 20B.
6. "Barycentric Refinement of a Barycentric Refinement is a Star Refinement". Mathematics Stack Exchange.
7. Brandsma, Henno (2003). "On paracompactness, full normality and the like" (PDF).

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.