Countably compact space: Difference between revisions
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==Definition== | ==Definition== | ||
===Symbol-free definition | ===Symbol-free definition=== | ||
A [[topological space]] is said to be '''countably compact''' if every countable open [[cover]] has a finite [[subcover]]. In other words, given a countable collection of open subsets whose union is the whole space, there is a finite subcollection whose union is again the whole space. | A [[topological space]] is said to be '''countably compact''' if every countable open [[cover]] has a finite [[subcover]]. In other words, given a countable collection of open subsets whose union is the whole space, there is a finite subcollection whose union is again the whole space. | ||
Revision as of 09:59, 20 August 2007
This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces
This is a variation of compactness. View other variations of compactness
Definition
Symbol-free definition
A topological space is said to be countably compact if every countable open cover has a finite subcover. In other words, given a countable collection of open subsets whose union is the whole space, there is a finite subcollection whose union is again the whole space.
Formalisms
Refinement formal expression
In the refinement formalism, a refinement formal expression is:
Countable open Finite open
viz, every countable open cover has a finite open refinement.
It is also an instance of the countably qualifier applied to compactness-like properties.