Star refinement

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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.

The general definition makes sense for arbitrary coverings and does not require a topology. Let X be a set and let \mathcal U=(U_i)_{i\in I} be a covering of X, i.e., X=\bigcup_{i\in I}U_i. Given a subset S of X then the star of S with respect to \mathcal U is the union of all the sets U_i that intersect S, i.e.:

\mathrm{st}(S,\mathcal U)=\bigcup\big\{U_i:i\in I,\ S\cap U_i\ne\emptyset\big\}.

Given a point x\in X, we write \mathrm{st}(x,\mathcal U) instead of \mathrm{st}(\{x\},\mathcal U).

The covering \mathcal U=(U_i)_{i\in I} of X is said to be a refinement of a covering \mathcal V=(V_j)_{j\in J} of X if every U_i is contained in some V_j. The covering \mathcal U is said to be a barycentric refinement of \mathcal V if for every x\in X the star \mathrm{st}(x,\mathcal U) is contained in some V_j. Finally, the covering \mathcal U is said to be a star refinement of \mathcal V if for every i\in I the star \mathrm{st}(U_i,\mathcal U) is contained in some V_j.

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.

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