mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, specifically in the study of

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

and

open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\s ...

s of a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...

''X'', a star refinement is a particular kind of

refinement of an open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\s ...

of ''X''. The general definition makes sense for arbitrary coverings and does not require a topology. Let

X

be a set and let

\mathcal U

be a covering of

X

, i.e.,

X = \bigcup \mathcal U

. Given a subset

S

X

then the ''star'' of

S

with respect to

\mathcal U

is the union of all the sets

U\in \mathcal U

that intersect

S

, i.e.: :

\operatorname(S, \mathcal U) = \bigcup\big\.

Given a point

x\in X

, we write

\operatorname(x,\mathcal U)

instead of

\operatorname(\, \mathcal U)

. Note that

\operatorname(S, \mathcal U) \ne \bigcup_ (O \cap S)

. The covering

\mathcal U

X

is said to be a ''refinement'' of a covering

\mathcal V

X

if every

U\in \mathcal U

is contained in some

V\in \mathcal V

. The covering

\mathcal U

is said to be a ''barycentric refinement'' of

\mathcal V

if for every

x\in X

the star

\operatorname(x,\mathcal U)

is contained in some

V\in\mathcal V

. Finally, the covering

\mathcal U

is said to be a ''star refinement'' of

\mathcal V

if for every

U\in \mathcal U

the star

\operatorname(U,\mathcal U)

is contained in some

V\in \mathcal V

. Star refinements are used in the definition of

fully normal space In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal, ...

and in one definition of

uniform space In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifo ...

. It is also useful for stating a characterization of paracompactness.

References

* J. Dugundji, Topology, Allyn and Bacon Inc., 1966. *

Lynn Arthur Steen Lynn Arthur Steen (January 1, 1941 – June 21, 2015) was an American mathematician who was a Professor of Mathematics at St. Olaf College, Northfield, Minnesota in the U.S. He wrote numerous books and articles on the teaching of mathematics. H ...

and

J. Arthur Seebach, Jr. J. Arthur Seebach Jr (May 17, 1938 – December 3, 1996) was an American mathematician. Seebach studied Greek language as an undergraduate, making it a second major with mathematics. Seebach studied with A. I. Weinzweig at Northwestern Univ ...

; 1970; ''

Counterexamples in Topology ''Counterexamples in Topology'' (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr. In the process of working on problems like the metrization problem, topologists (including Steen and Seebach) hav ...

''; 2nd (1995) Dover edition {{ISBN, 0-486-68735-X; page 165. Topology