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 $X$ be a set and let $\mathcal U$ be a covering of $X$, i.e., $X = \bigcup \mathcal U$. Given a subset $S$ of $X$ 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 $\mathcal U$ of $X$ is said to be a ''refinement'' of a covering $\mathcal V$ of $X$ 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 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.

Topology