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.
be a set and let
is the union of all the sets
st (
{\displaystyle \operatorname {st} (S,{\mathcal {U}})=\bigcup {\big \{}U\in {\mathcal {U}}:S\cap U\neq \varnothing {\big \}}.}
st ( x ,
{\displaystyle \operatorname {st} (x,{\mathcal {U}})}
st ( { x } ,
{\displaystyle \operatorname {st} (\{x\},{\mathcal {U}}).}
is a refinement of a covering
The following are two special kinds of refinement.
is called a barycentric refinement of
st ( x ,
{\displaystyle \operatorname {st} (x,{\mathcal {U}})}
is called a star refinement of
st (
{\displaystyle \operatorname {st} (U,{\mathcal {U}})}
[3][2] 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.
[4][5][6][7] Given a metric space
be the collection of all open balls
of a fixed radius
is a barycentric refinement of
is a star refinement of