Editing Star refinement (section)

==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>