Cover

Let $𝑋$ be a set A collection $U \subseteq T$ of subsets of $𝑋$ is called a cover iff $𝑋 \subseteq ⋃ 𝑈 \in U 𝑈$ . #m/def/topology Typically $𝑋$ is a topological space, in which case $U$ is called an open cover iff every $𝑈 \in U$ is open.

Further terminology

A subcover of $U$ is a a subcollection of $U$ that is also a cover of $𝑋$ .
A refinement of $U$ is a cover $V$ such that every $𝑉 \in V$ is contained in at least one $𝑈 \in U$ , i.e. $𝑈 \subseteq 𝑉$ .
A cover $U$ is locally finite iff every $𝑥 \in 𝑋$ as a neighbourhood intersecting with finitely many $𝑈 \in U$ .

Properties

A space is compact iff every open cover has a finite subcover.
A space is paracompact iff every open cover has a locally finite open refinement.

#state/tidy | #lang/en | #SemBr