Compactness axioms: Difference between revisions

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imported>Richard Pinch
(→‎Definitions: added Locally compact, Strongly locally compact, σ-locally compact, Pseudocompact)
imported>Richard Pinch
(added definition and anchor: point finite cover)
 
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In [[general topology]], the important property of '''[[compact set|compactness]]''' has a number of related properties.
In [[general topology]], the important property of '''[[compact set|compactness]]''' has a number of related properties.


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

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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.