Compact space: Difference between revisions
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==Definition== | |||
A topological space is termed a '''compact space''' if it satisfies the following equivalent conditions: | |||
* Every open cover of the topological space has a finite subcover. | |||
* If a collection of closed subsets of the space has the property that every finite subcollection has a nonempty intersection, then the whole collection has a nonempty intersection. | |||
* The only [[maximal ideal]]s in the [[ring of continuous real-valued functions on a topological space|ring of continuous real-valued functions]] on the topological space are the ideals of functions vanishing at a particular point. | |||
===Equivalence of definitions=== | |||
The third definition is equivalent to the first two because: [[Natural map from topological space to max-spectrum of ring of continuous real-valued functions is a surjection iff the space is compact]]. | |||
==External links== | |||
===Primary subject wiki entry=== | |||
[[Topospaces:Compact space]] | |||
Latest revision as of 00:11, 6 February 2009
Definition
A topological space is termed a compact space if it satisfies the following equivalent conditions:
- Every open cover of the topological space has a finite subcover.
- If a collection of closed subsets of the space has the property that every finite subcollection has a nonempty intersection, then the whole collection has a nonempty intersection.
- The only maximal ideals in the ring of continuous real-valued functions on the topological space are the ideals of functions vanishing at a particular point.
Equivalence of definitions
The third definition is equivalent to the first two because: Natural map from topological space to max-spectrum of ring of continuous real-valued functions is a surjection iff the space is compact.