# Definition:Open Cover

## Definition

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $\mathcal C$ be a cover for $S$.

Then $\mathcal C$ is an open cover (of $T$) if and only if:

$\mathcal C \subseteq \tau$

That is, if and only if all the elements of $\mathcal C$ are open sets.

### Open Cover of Subset

Let $H$ be a subset of $S$.

Let $\mathcal C$ be a cover of $H$.

Then $\mathcal C$ is an open cover for $H$ if and only if:

$\mathcal C \subseteq \tau$

That is, if and only if all the elements of $\mathcal C$ are open sets.

## Also known as

Some sources have this as open covering.