Editing Star refinement (section)

==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}}