Editing Star refinement

In [[mathematics]], specifically in the study of [[topology]] and [[open cover]]s 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 <math>X</math> be a set and let <math>\mathcal U</math> be a [[cover (topology)|covering]] of <math>X,</math> that is, <math display="inline">X = \bigcup \mathcal U.</math> Given a subset <math>S</math> of <math>X,</math> the '''star''' of <math>S</math> with respect to <math>\mathcal U</math> is the union of all the sets <math>U \in \mathcal U</math> that intersect <math>S,</math> that is,
<math display=block>\operatorname{st}(S, \mathcal U) = \bigcup\big\{U \in \mathcal U: S\cap U \neq \varnothing\big\}.</math>

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

A covering <math>\mathcal U</math> of <math>X</math> is a [[refinement (topology)|refinement]] of a covering <math>\mathcal V</math> of <math>X</math> if every <math>U \in \mathcal U</math> is contained in some <math>V \in \mathcal V.</math>  The following are two special kinds of refinement.  The covering <math>\mathcal U</math> is called a '''barycentric refinement''' of <math>\mathcal V</math> if for every <math>x \in X</math> the star <math>\operatorname{st}(x,\mathcal U)</math> is contained in some <math>V \in \mathcal V.</math>{{sfn|Dugundji|1966|loc=Definition VIII.3.1, p. 167}}{{sfn|Willard|2004|loc=Definition 20.1}}  The covering <math>\mathcal U</math> is called a '''star refinement''' of <math>\mathcal V</math> if for every <math>U \in \mathcal U</math> the star <math>\operatorname{st}(U, \mathcal U)</math> is contained in some <math>V \in \mathcal V.</math>{{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}}<ref>{{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}}</ref><ref>{{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}}</ref>

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

==See also==

* {{annotated link|Family of sets}}

==Notes==
{{reflist}}

==References==

* {{Dugundji Topology}} <!--{{sfn|Dugundji|1966|p=}}-->
* {{Willard General Topology}}  <!--{{sfn|Willard|2004|p=}}-->

[[Category:General topology]]