# Limit point-compact space

From Topospaces

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

## Contents

## 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.