Definition:Subcover

Definition

Let $S$ be a set.

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

A subcover of $\UU$ for $S$ is a set $\VV \subseteq \UU$ such that $\VV$ is also a cover for $S$.

Finite Subcover

A finite subcover of $\UU$ for $S$ is a subcover $\VV \subseteq \UU$ which is finite.

Countable Subcover

A countable subcover of $\UU$ for $S$ is a subcover $\VV \subseteq \UU$ which is countable.

Also see

• Definition:Refinement of Cover

• Results about subcovers can be found here.

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $5$: Compact spaces: $5.2$: Definition of compactness: Definitions $5.2.1$