Compact metrizable space: Difference between revisions
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A [[topological space]] is termed '''compact metrizable''' if it is [[compact space|compact]] and [[metrizable space|metrizable]], or eqiuvalently, if it can be given the structure of a [[compact metric space]]. | A [[topological space]] is termed '''compact metrizable''' if it is [[compact space|compact]] and [[metrizable space|metrizable]], or eqiuvalently, if it can be given the structure of a [[compact metric space]]. | ||
{{topospace property}} | |||
{{topospace property conjunction|compact space|metrizable space}} | {{topospace property conjunction|compact space|metrizable space}} | ||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ||
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| [[Stronger than::supercompact space]]|| || || || | | [[Stronger than::supercompact space]]|| there is a [[subbasis]] of [[open subset]]s such that any [[open cover]] using that subbasis has a subcover using at most two subsets || || || | ||
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| [[Stronger than::compact space]] || || || || {{intermediate notions short|compact space|compact metrizable space}} | | [[Stronger than::compact space]] || || || || {{intermediate notions short|compact space|compact metrizable space}} | ||
Latest revision as of 18:44, 28 January 2012
Definition
A topological space is termed compact metrizable if it is compact and metrizable, or eqiuvalently, if it can be given the structure of a compact metric space.
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 article describes a property of topological spaces obtained as a conjunction of the following two properties: compact space and metrizable space
Relation with other properties
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| supercompact space | there is a subbasis of open subsets such that any open cover using that subbasis has a subcover using at most two subsets | |||
| compact space | Compact Hausdorff space|FULL LIST, MORE INFO | |||
| completely metrizable space | ||||
| compact Hausdorff space | |FULL LIST, MORE INFO |