|
|
|
|
|
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
- If $X$ is a set, then $\{X\}$ is a cover of $X$ .
- The power set of a set $X$ is a cover of $X$ .
- A topology for a set is a cover of that set.
- 1
- J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
|
Anyone with an account can edit this entry. Please help improve it!
"cover" is owned by mps. [ full author list (6) | owner history (2) ]
|
|
(view preamble | get metadata)
See Also: compact, -net, site, covering space, a compact metric space is second countable
| Also defines: |
open cover, subcover, refinement, finite cover, countable cover, uncountable cover, open subcover, open refinement, cover refinement |
| Keywords: |
topology, set theory |
|
|
Cross-references: power set, open sets, topological space, preorder, uncountable, countable, finite, subset
There are 89 references to this entry.
This is version 15 of cover, born on 2002-01-04, modified 2009-01-01.
Object id is 1224, canonical name is Cover.
Accessed 20107 times total.
Classification:
| AMS MSC: | 54A99 (General topology :: Generalities :: Miscellaneous) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|