# cover

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 $\mathcal{U}^{\prime}\subset\mathcal{U}$ such that $\mathcal{U}^{\prime}$ 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. 1.

If $X$ is a set, then $\{X\}$ is a cover of $X$.

2. 2.

The power set of a set $X$ is a cover of $X$.

3. 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.
 Title cover Canonical name Cover Date of creation 2013-03-22 12:06:31 Last modified on 2013-03-22 12:06:31 Owner mps (409) Last modified by mps (409) Numerical id 19 Author mps (409) Entry type Definition Classification msc 54A99 Related topic Compact Related topic VarepsilonNet Related topic Site Related topic CoveringSpace Related topic CompactMetricSpacesAreSecondCountable Defines open cover Defines subcover Defines refinement Defines finite cover Defines countable cover Defines uncountable cover Defines open subcover Defines open refinement Defines cover refinement