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.

Given x in X and an open cover {U_i} of X with index set I, the star of x with respect to the cover is the set of i in I such that x is in U_i. That is,

For each x we can take the union of the sets in its star; this is also called the star of x. That is,

Then this open cover is a star refinement of some other open cover {V_j}, with index set J, if

1. each U_i is contained in some V_j (the refinement condition), and
2. each star U^*(x) is contained in a V_j (star condition).

That is,

1. 
2. 

Actually, the star condition alone is enough; the refinement condition follows (except in a degenerate case when X is the empty set).

Star refinements are used in the definition of fully normal space and in one definition of uniform space.

[edit] References

• Lynn Arthur Steen and J. Arthur Seebach, Jr.; 1970; Counterexamples in Topology; 2nd (1995) Dover edition ISBN 048668735X; page 165.