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 general definition makes sense for arbitrary coverings and does not require a topology. Let be a set and let
be a covering of
, i.e.,
. Given a subset
of
then the star of
with respect to
is the union of all the sets
that intersect
, i.e.:
Given a point , we write
instead of
.
The covering of
is said to be a refinement of a covering
of
if every
is contained in some
. The covering
is said to be a barycentric refinement of
if for every
the star
is contained in some
. Finally, the covering
is said to be a star refinement of
if for every
the star
is contained in some
.
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.
References
- J. Dugundji, Topology, Allyn and Bacon Inc., 1966.
- Lynn Arthur Steen and J. Arthur Seebach, Jr.; 1970; Counterexamples in Topology; 2nd (1995) Dover edition ISBN 0-486-68735-X; page 165.