# 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**.

## Examples

### Arbitrary Example

Let $S = \set {1, 2, 3, 4, 5, 6}$.

Let $H \subseteq S = \set {1, 3}$.

Let $\CC = \set {\set {1, 2}, \set {3, 4} }$.

Then $\CC$ is a cover of $H$.

## Also see

- Results about
**covers**can be found**here**.