# Definition:Cover of Set

## Definition

Let $S$ be a set.

### Definition 1

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

$\ds S \subseteq \bigcup \CC$

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

### Definition 2

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

$\forall s \in S : \exists C \in \CC : x \in C$

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 $\CC$ for $S$ is a countable cover if and only if $\CC$ is a countable set.

### $\delta$-Cover

Let $\struct {S, d}$ be a metric space.

Let $\delta \in \R_{>0}$.

A cover $\CC$ for $S$ is a $\delta$-cover if and only if $\CC$ is a countable cover such that:

$\forall U \in \CC :0 < \size U \le \delta$

where $\size U$ denotes the diameter of $U$.

## Also known as

A cover is also known as a covering.

## Also see

• Results about covers can be found here.