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.:
![{\displaystyle \operatorname {st} (S,{\mathcal {U}})=\bigcup {\big \{}U:U\in {\mathcal {U}},\ S\cap U\neq \emptyset {\big \}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1dc1d256359e54abea07745c173cbb61322c87c)
Given a point
, we write
instead of
. Note that
.
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