Definition:Refinement of Cover
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Definition
Let $S$ be a set.
Let $\UU = \set {U_\alpha}$ and $\VV = \set {V_\beta}$ be covers of $S$.
Then $\VV$ is a refinement of $\UU$ if and only if:
- $\forall V_\beta \in \VV: \exists U_\alpha \in \UU: V_\beta \subseteq U_\alpha$
That is, if and only if every element of $\VV$ is the subset of some element of $\UU$.
Finer Cover
Let $\VV$ be a refinement of $\UU$.
Then $\VV$ is finer than $\UU$.
Coarser Cover
Let $\VV$ be a refinement of $\UU$.
Then $\UU$ is coarser than $\VV$.
Note
Although specified for the cover of a set, a refinement is usually used in the context of a topological space.
Also see
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