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.
Famous quotes containing the words star and/or refinement:
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That slip from heaven at night and softly fall
Has been picked up with stones to build a wall.”
—Robert Frost (1874–1963)
“It is an immense loss to have all robust and sustaining expletives refined away from one! At ... moments of trial refinement is a feeble reed to lean upon.”
—Alice James (1848–1892)