Difference between revisions of "Relatively-compact set"
From Encyclopedia of Mathematics
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| − | {{ | + | A subset $A$ of a [[topological space]] $X$ with the property that the [[Closure of a set|closure]] $\bar A$ of $A$ in $X$ is [[Compact space|compact]]. |
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| + | A subset $A$ of a metric space $X$ is relatively compact if and only if every sequence of points in $A$ has a cluster point in $X$. | ||
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| + | A space is compact if it is relatively compact in itself. | ||
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| + | An alternative definition is that $A$ is relatively compact in $X$ if and only if every open cover of $X$ contains a finite subcover of $A$. This formulation is equivalent to requiring that the set $A$ be [[way below]] $X$ with respect to set inclusion and the directed set of open subsets of $X$. | ||
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| + | ====References==== | ||
| + | * N. Bourbaki, "General Topology" Volume 4 Ch.5-10, Springer [1974] (2007) ISBN 3-540-34399-7 {{ZBL|1107.54002}} | ||
| + | * G. Gierz, Karl Heinrich Hofmann, K. Keimel, J.D. Lawson, M. Mislove, Dana S. Scott, "A compendium of continuous lattices" Springer (1980) ISBN 3-540-10111-X {{MR|0614752}} {{ZBL|0452.06001}} | ||
Revision as of 19:17, 20 January 2021
2020 Mathematics Subject Classification: Primary: 54D30 [MSN][ZBL]
A subset $A$ of a topological space $X$ with the property that the closure $\bar A$ of $A$ in $X$ is compact.
A subset $A$ of a metric space $X$ is relatively compact if and only if every sequence of points in $A$ has a cluster point in $X$.
A space is compact if it is relatively compact in itself.
An alternative definition is that $A$ is relatively compact in $X$ if and only if every open cover of $X$ contains a finite subcover of $A$. This formulation is equivalent to requiring that the set $A$ be way below $X$ with respect to set inclusion and the directed set of open subsets of $X$.
References
- N. Bourbaki, "General Topology" Volume 4 Ch.5-10, Springer [1974] (2007) ISBN 3-540-34399-7 Zbl 1107.54002
- G. Gierz, Karl Heinrich Hofmann, K. Keimel, J.D. Lawson, M. Mislove, Dana S. Scott, "A compendium of continuous lattices" Springer (1980) ISBN 3-540-10111-X MR0614752 Zbl 0452.06001
How to Cite This Entry:
Relatively-compact set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Relatively-compact_set&oldid=42456
Relatively-compact set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Relatively-compact_set&oldid=42456
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article