Compactness axioms

In general topology, the important property of compactness has a number of related properties.

We say that a topological space X is
 * Compact if every cover by open sets has a finite subcover.
 * Countably compact if every countable cover by open sets has a finite subcover.
 * Lindelöf if every cover by open sets has a countable subcover.
 * Sequentially compact if every convergent sequence has a convergent subsequence.
 * Paracompact if every cover by open sets has an open locally finite refinement.
 * Metacompact if every cover by open sets has a point finite open refinement.
 * Orthocompact if every cover by open sets has an interior preserving open refinement.
 * σ-compact if it is the union of countably many compact subspaces.