Definition:Precise Refinement of Cover
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Definition
Let $\family{S_i}_{i \in I}$ be an indexed family of subsets of a set $S$ indexed by $I$.
Let:
- $\SS = \set{S_i: i \in I}$
be a cover of $S$.
Also let $\family{T_i}_{i \in I}$ be an indexed family of subsets of $S$ indexed by $I$.
Let:
- $\TT = \set {T_i: i \in I}$
be a cover of $S$.
Then $\TT$ is a precise refinement of $\SS$ if and only if:
- $\forall i \in I: T_i \subseteq S_i$
Sources
- 2000: James R. Munkres: Topology (2nd ed.): $6$: Metrization Theorems and Paracompactness: $\S 41$: Paracompactness: Lemma $41.6$