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 term has two similar but distinct usages. A related term sometimes used to differentiate the weaker of these two properties is the notion of a barycentric refinement.

Star refinements are used in the definition of a fully normal space, and in one among several equivalent formulations of a uniform space.

Definitions

be a covering the star

is a refinement is called a barycentric refinement is called a star refinement

is called fully normal has a barycentric open refinement.

Properties and Examples

Every star refinement of a cover is a barycentric refinement of that cover. The converse is not true, but a barycentric refinement of a barycentric refinement is a star refinement.

Given a metric space

By a theorem of A.H. Stone, for a T1 space being fully normal and being paracompact are equivalent. This was a landmark theorem and provided the first proof that metric spaces are paracompact. The proof is difficult, but simpler proofs of the paracompactness of metric spaces were later provided.

See also

Notes

References