# Compactness axioms

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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 such that the union is equal to *X*. A *subcover* is a subfamily which is again a cover where *B* is a subset of *A*. A *refinement* is a cover such that for each β in *B* there is an α in *A* such that . 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 "open cover" 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. **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.**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.