In mathematics, specifically in the study of

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...

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 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 po ...

''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\ ...

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 un ...

. 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 ...

and J. Arthur Seebach, Jr.; 1970; '' Counterexamples in Topology''; 2nd (1995) Dover edition {{ISBN, 0-486-68735-X; page 165. Topology