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 $X$ be a set and let $\mathcal U$ be a covering of $X,$ that is, $X = \bigcup \mathcal U.$ 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 $V \in \mathcal V.$ {{sfn|Dugundji|1966|loc=Definition VIII.3.1, p. 167}}{{sfn|Willard|2004|loc=Definition 20.1}} 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 $V \in \mathcal V.$ {{sfn|Dugundji|1966|loc=Definition VIII.3.3, p. 167}}{{sfn|Willard|2004|loc=Definition 20.1}}

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.{{sfn|Dugundji|1966|loc=Prop. VIII.3.4, p. 167}}{{sfn|Willard|2004|loc=Problem 20B}}{{cite web |title=Barycentric Refinement of a Barycentric Refinement is a Star Refinement |url=https://math.stackexchange.com/questions/3168765 |website=Mathematics Stack Exchange |language=en}}{{cite web |last1=Brandsma |first1=Henno |title=On paracompactness, full normality and the like |date=2003 |url=http://at.yorku.ca/p/a/c/a/02.pdf}}

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 $\mathcal V.$

Notes

References

{{Dugundji Topology}}
{{Willard General Topology}}

Category:General topology

star refinement

Definitions

Properties and Examples

See also

Notes

References