cover

Definition ([1], pp. 49) Let $Y$ be a subset of a set $X$. A cover for $Y$ is a collection of 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 be finite, countable, or uncountable. The cover is correspondingly called a finite 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.

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