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Definition ([1], pp. 49) Let  be a subset of a set  . A cover for  is a collection of sets
 such that each  is a subset of  , and
The collection of sets can be arbitrary, that is,  can be finite, countable, or uncountable. The cover is correspondingly called a finite cover, countable cover, or uncountable cover.
A subcover of
 is a subset
 such that
 is also a cover of  .
A refinement
 of
 is a cover of  such that for every
 there is some
 such that
 . When
 refines
 , it is usually written
 .  is a preorder on the set of covers of any topological space  .
If  is a topological space and the members of
 are open sets, then
 is said to be an open cover. Open subcovers and open refinements are defined similarly.
Examples
- If
 is a set, then  is a cover of  .
- The power set of a set
 is a cover of  .
- A topology for a set is a cover of that set.
- 1
- J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
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"cover" is owned by mps. [ full author list (6) | owner history (2) ]
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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 |
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Cross-references: power set, open sets, topological space, preorder, uncountable, countable, finite, subset
There are 81 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 18669 times total.
Classification:
| AMS MSC: | 54A99 (General topology :: Generalities :: Miscellaneous) |
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Pending Errata and Addenda
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