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

• Equivalence of Definitions of Cover of Set

• Definition:Open Cover

• Results about covers can be found here.