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 be a set and let be a covering of , i.e., . Given a subset of then the star of with respect to is the union of all the sets that intersect , i.e.:

Given a point , we write instead of .

The covering of is said to be a refinement of a covering of if every is contained in some . The covering is said to be a barycentric refinement of if for every the star is contained in some . Finally, the covering is said to be a star refinement of if for every the star is contained in some .

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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