cover
Definition ([1], pp. 49) Let $Y$ be a subset of a set $X$. A cover for $Y$ is a collection^{} of sets $\mathrm{\u0111\x9d\x92\xb0}={\{{U}_{i}\}}_{i\xe2\x88\x88I}$ such that each ${U}_{i}$ is a subset of $X$, and
$$Y\xe2\x8a\x82\underset{i\xe2\x88\x88I}{\xe2\x8b\x83}{U}_{i}.$$ |
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 $\mathrm{\u0111\x9d\x92\xb0}$ is a subset ${\mathrm{\u0111\x9d\x92\xb0}}^{\xe2\x80\u010c}\xe2\x8a\x82\mathrm{\u0111\x9d\x92\xb0}$ such that ${\mathrm{\u0111\x9d\x92\xb0}}^{\xe2\x80\u010c}$ is also a cover of $X$.
A refinement $\mathrm{\u0111\x9d\x92\pm}$ of $\mathrm{\u0111\x9d\x92\xb0}$ is a cover of $X$ such that for every $V\xe2\x88\x88\mathrm{\u0111\x9d\x92\pm}$ there is some $U\xe2\x88\x88\mathrm{\u0111\x9d\x92\xb0}$ such that $V\xe2\x8a\x82U$. When $\mathrm{\u0111\x9d\x92\pm}$ refines $\mathrm{\u0111\x9d\x92\xb0}$, it is usually written $\mathrm{\u0111\x9d\x92\pm}\xe2\u0218\u017b\mathrm{\u0111\x9d\x92\xb0}$. $\xe2\u0218\u017b$ is a preorder on the set of covers of any topological space^{} $X$.
If $X$ is a topological space and the members of $\mathrm{\u0111\x9d\x92\xb0}$ are open sets, then $\mathrm{\u0111\x9d\x92\xb0}$ is said to be an open cover. Open subcovers and open refinements are defined similarly.
Examples
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 |