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