Compactness axioms: Difference between revisions
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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== | ||
We say that a [[topological space]] ''X'' is | We say that a [[topological space]] ''X'' is | ||
* '''Compact''' if every cover by [[open set]]s has a finite subcover. | * '''Compact''' if every cover by [[open set]]s has a finite subcover. | ||
* A '''compactum''' if it is a compact [[metric space]]. | |||
* '''Countably compact''' if every [[countable set|countable]] cover by open sets has a finite subcover. | * '''Countably compact''' if every [[countable set|countable]] cover by open sets has a finite subcover. | ||
* '''Lindelöf''' if every cover by open sets has a countable subcover. | * '''Lindelöf''' if every cover by open sets has a countable subcover. | ||
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* '''Orthocompact''' if every cover by open sets has an interior preserving 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. | * '''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 function|continuous]] [[real number|real]]-valued [[function (mathematics)|function]] is bounded. | ||
Latest revision as of 07:47, 30 December 2008
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.