Limit point-compact space

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.

Stronger properties

 * Compact space
 * Countably compact space
 * Sequentially compact space

Metaproperties
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.