Limit point-compact space

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

• Compact space
• Countably compact space
• Sequentially compact space

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.