Cover (topology) - Compactness

The language of covers is often used to define several topological properties related to compactness. A topological space X is said to be

• Compact, if every open cover has a finite subcover, (or equivalently that every open cover has a finite refinement);
• Lindelöf, if every open cover has a countable subcover, (or equivalently that every open cover has a countable refinement);
• Metacompact, if every open cover has a point finite open refinement;
• Paracompact, if every open cover admits a locally finite open refinement.

For some more variations see the above articles.