Compactness axioms

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

The definitions require some preliminary terminology. A cover of a set X is a family $$\mathcal{U} = \{ U_\alpha : \alpha \in A \}$$ such that the union $$\bigcup_{\alpha \in A} U_\alpha$$ is equal to X. A  is a subfamily which is again a cover $$\mathcal{S} = \{ U_\alpha : \alpha \in B \}$$ where B is a subset of A. A refinement is a cover $$\mathcal{R} = \{ V_\beta : \beta \in B \}$$ such that for each β in B there is an α in A such that $$V_\beta \subseteq U_\alpha$$. A cover is finite or countable if the index set is finite or countable. A cover is point finite if each element of X belongs to a finite numbers of sets in the cover. The phrase "" is often used to denote "cover by open sets".

Definitions
We say that a topological space X is
 * Compact if every cover by open sets has a finite subcover.
 * A compactum if it is a compact metric space.
 *  if every countable cover by open sets has a finite subcover.
 * Lindelöf if every cover by open sets has a countable subcover.
 *  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.
 * Locally compact if every point has a compact neighbourhood.
 * Strongly locally compact if every point has a neighbourhood with compact closure.
 * σ-locally compact if it is both σ-compact and locally compact.
 * Pseudocompact if every continuous real-valued function is bounded.