Definition:Cover of Set

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Definition

Let $S$ be a set.


A cover for $S$ is a set of sets $\mathcal C$ such that:

$\displaystyle S \subseteq \bigcup \mathcal C$

where $\bigcup \mathcal C$ denotes the union of $\mathcal C$.


We say that $S$ is covered by $\mathcal C$.


Cover of Subset

Let $A \subseteq S$ be a subset.

Let $\mathcal C$ be a set of subsets of $S$.


Then $\mathcal C$ is a cover of $A$ if and only if $A \subseteq \displaystyle \bigcup \mathcal C$, where $\cup$ denotes union.


Finite Cover

A cover $\mathcal C$ for $S$ is a finite cover if and only if $\mathcal C$ is a finite set.


Countable Cover

A cover $\mathcal C$ for $S$ is a countable cover if and only if $\mathcal C$ is a countable set.


Also known as

A cover is also known as a covering.


Also see

  • Results about covers can be found here.


Sources