Definition:Open Cover

Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $\CC$ be a cover for $S$.

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

$\CC \subseteq \tau$

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

Open Cover of Subset

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

Let $\CC$ be a cover of $H$.

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

$\CC \subseteq \tau$

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

Also known as

Some sources have this as open covering.

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $5$: Compact spaces: $5.2$: Definition of compactness: Definitions $5.2.1$
• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): open cover
• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 1.5$: Normed and Banach spaces. Compact sets