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]
Properties and Examplesedit
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
See alsoedit
Family of sets – Any collection of sets, or subsets of a set
star, refinement, mathematics, specifically, study, topology, open, covers, topological, space, star, refinement, particular, kind, refinement, open, cover, related, concept, notion, barycentric, refinement, used, definition, fully, normal, space, definition, . 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 A related concept is the notion of barycentric refinement 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 Contents 1 Definitions 2 Properties and Examples 3 See also 4 Notes 5 ReferencesDefinitions editThe general definition makes sense for arbitrary coverings and does not require a topology Let X displaystyle X nbsp be a set and let U displaystyle mathcal U nbsp be a covering of X displaystyle X nbsp that is X U textstyle X bigcup mathcal U nbsp Given a subset S displaystyle S nbsp of X displaystyle X nbsp the star of S displaystyle S nbsp with respect to U displaystyle mathcal U nbsp is the union of all the sets U U displaystyle U in mathcal U nbsp that intersect S displaystyle S nbsp that is st S U U U S U displaystyle operatorname st S mathcal U bigcup big U in mathcal U S cap U neq varnothing big nbsp Given a point x X displaystyle x in X nbsp we write st x U displaystyle operatorname st x mathcal U nbsp instead of st x U displaystyle operatorname st x mathcal U nbsp A covering U displaystyle mathcal U nbsp of X displaystyle X nbsp is a refinement of a covering V displaystyle mathcal V nbsp of X displaystyle X nbsp if every U U displaystyle U in mathcal U nbsp is contained in some V V displaystyle V in mathcal V nbsp The following are two special kinds of refinement The covering U displaystyle mathcal U nbsp is called a barycentric refinement of V displaystyle mathcal V nbsp if for every x X displaystyle x in X nbsp the star st x U displaystyle operatorname st x mathcal U nbsp is contained in some V V displaystyle V in mathcal V nbsp 1 2 The covering U displaystyle mathcal U nbsp is called a star refinement of V displaystyle mathcal V nbsp if for every U U displaystyle U in mathcal U nbsp the star st U U displaystyle operatorname st U mathcal U nbsp is contained in some V V displaystyle V in mathcal V nbsp 3 2 Properties and Examples editEvery 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 X displaystyle X nbsp let V Bϵ x x X displaystyle mathcal V B epsilon x x in X nbsp be the collection of all open balls Bϵ x displaystyle B epsilon x nbsp of a fixed radius ϵ gt 0 displaystyle epsilon gt 0 nbsp The collection U Bϵ 2 x x X displaystyle mathcal U B epsilon 2 x x in X nbsp is a barycentric refinement of V displaystyle mathcal V nbsp and the collection W Bϵ 3 x x X displaystyle mathcal W B epsilon 3 x x in X nbsp is a star refinement of V displaystyle mathcal V nbsp See also editFamily of sets Any collection of sets or subsets of a setNotes edit Dugundji 1966 Definition VIII 3 1 p 167 a b 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 editDugundji 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 Retrieved from https en wikipedia org w index php title Star refinement amp oldid 1171272591, wikipedia, wiki, book, books, library,