Definition:Star Refinement
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Definition
Let $S$ be a set.
Let $\CC$ be a cover for $S$.
Let $\VV$ be a cover for $S$ such that:
- $ \forall x \in S: \exists U \in \CC: x^* \subseteq U$
where $x^*$ is the star of $x$ with respect to $\VV$.
That is:
- $\ds x^* := \bigcup \set {V \in \VV: x \in V}$
the union of all sets in $\VV$ which contain $x$.
Then $\VV$ is a star refinement of $\CC$.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Paracompactness