Definition:Cover of Set

From ProofWiki
Jump to navigation Jump to search

Definition

Let $S$ be a set.


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

$\ds S \subseteq \bigcup \CC$

where $\bigcup \CC$ denotes the union of $\CC$.


We say that $S$ is covered by $\CC$.


Cover of Subset

Let $T \subseteq S$ be a subset.

Let $\CC$ be a set of subsets of $S$.


Then $\CC$ is a cover of $T$ if and only if $T \subseteq \ds \bigcup \CC$, where $\cup$ denotes union.


Finite Cover

A cover $\CC$ for $S$ is a finite cover if and only if $\CC$ 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