Definition ([1], pp. 49) Let $Y$ be a subset of a set $X$ . A cover for $Y$ is a collection of sets $\mathcal{U}=\{U_i\}_{i\in I}$ such that each $U_i$ is a subset of $X$ , and $$ Y \subset \bigcup_{i\in I} U_i.$$ The collection of sets can be arbitrary, that is, $I$ can be finite, countable, or uncountable. The cover is correspondingly called a finite cover, countable cover, or uncountable cover.

A subcover of $\mathcal{U}$ is a subset $\mathcal{U}'\subset\mathcal{U}$ such that $\mathcal{U}'$ is also a cover of $X$ .

A refinement $\mathcal{V}$ of $\mathcal{U}$ is a cover of $X$ such that for every $V\in\mathcal{V}$ there is some $U\in\mathcal{U}$ such that $V\subset U$ . When $\mathcal{V}$ refines $\mathcal{U}$ , it is usually written $\mathcal{V}\preceq \mathcal{U}$ . $\preceq$ is a preorder on the set of covers of any topological space $X$ .

If $X$ is a topological space and the members of $\mathcal{U}$ are open sets, then $\mathcal{U}$ is said to be an open cover. Open subcovers and open refinements are defined similarly.

Examples

1. If $X$ is a set, then $\{X\}$ is a cover of $X$ .
2. The power set of a set $X$ is a cover of $X$ .
3. A topology for a set is a cover of that set.

References

1

J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.