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Concept# Star refinement

Summary

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.
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 The covering is called a star refinement of if for every the star is contained in some
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.

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Related concepts (3)

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. 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. The general definition makes sense for arbitrary coverings and does not require a topology.

Metric space

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane.

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, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness.