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:
“What is this flesh I purchased with my pains,
This fallen star my milk sustains,
This love that makes my heart’s blood stop
Or strikes a sudden chill into my bones
And bids my hair stand up?”
—William Butler Yeats (1865–1939)
“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)