cover

Definition ([1], pp. 49)Let $Y$ be a subset of a set $X$. A cover for $Y$ is a collectionof sets $𝒰={\{U_i\}}_{i\in I}$ 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 befinite, countable, or uncountable. The cover is correspondingly called afinite cover, countable cover, or uncountable cover.

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

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

If $X$ is a topological space and the members of $𝒰$ are open sets,then $𝒰$ 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.