Star refinement
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
[edit]The general definition makes sense for arbitrary coverings and does not require a topology. Let be a set and let be a covering of that is, Given a subset of the star of with respect to is the union of all the sets that intersect that is,
Given a point we write instead of
A covering of is a refinement of a covering of if every is contained in some The following are two special kinds of refinement. The covering is called a barycentric refinement of if for every the star is contained in some [1][2] The covering is called a star refinement of if for every the star is contained in some [3][2]
A space is called fully normal if every open cover of has a barycentric open refinement.
Properties and Examples
[edit]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.[4][5][6][7]
Given a metric space let be the collection of all open balls of a fixed radius The collection is a barycentric refinement of and the collection is a star refinement of
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
[edit]- Family of sets – Any collection of sets, or subsets of a set
Notes
[edit]- ↑ Dugundji 1966, Definition VIII.3.1, p. 167.
- 1 2 Willard 2004, Definition 20.1.
- ↑ Dugundji 1966, Definition VIII.3.3, p. 167.
- ↑ Dugundji 1966, Prop. VIII.3.4, p. 167.
- ↑ Willard 2004, Problem 20B.
- ↑ "Barycentric Refinement of a Barycentric Refinement is a Star Refinement". Mathematics Stack Exchange.
- ↑ Brandsma, Henno (2003). "On paracompactness, full normality and the like" (PDF).
References
[edit]- Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485.
- Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.