Limit point-compact space: Difference between revisions
No edit summary |
m (5 revisions) |
||
| (4 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
{{topospace property}} | {{topospace property}} | ||
{{variationof|compactness}} | |||
==Definition== | ==Definition== | ||
| Line 5: | Line 7: | ||
===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[topological space]] is said to be '''limit point-compact''' if every infinite subset of it has a [[limit point]]. | A [[topological space]] is said to be '''limit point-compact''' or '''weakly countably compact''' if every infinite subset of it has a [[limit point]]. | ||
==Relation with other properties== | ==Relation with other properties== | ||
| Line 13: | Line 15: | ||
* [[Compact space]] | * [[Compact space]] | ||
* [[Countably compact space]] | * [[Countably compact space]] | ||
* [[Sequentially compact space]] | |||
==Metaproperties== | |||
{{coarsening-preserved}} | |||
If we switch to a coarser topology, whatever were earlier limit points of a set, continue to remain limit points (more may get added). Thus, the property of being limit point-compact is preserved upon switching to a coarser topology. | |||
Latest revision as of 19:48, 11 May 2008
This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces
This is a variation of compactness. View other variations of compactness
Definition
Symbol-free definition
A topological space is said to be limit point-compact or weakly countably compact if every infinite subset of it has a limit point.
Relation with other properties
Stronger properties
Metaproperties
Coarsening
This property of topological spaces is preserved under coarsening, viz, if a set with a given topology has the property, the same set with a coarser topology also has the property
If we switch to a coarser topology, whatever were earlier limit points of a set, continue to remain limit points (more may get added). Thus, the property of being limit point-compact is preserved upon switching to a coarser topology.