Compactly generated space: Difference between revisions
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==Definition== | ==Definition== | ||
===Symbol-free definition=== | |||
A [[topological space]] is said to be '''compactly generated''' if the topology on it is generated by a collection of compact subsets. In other words, a set in the topological space is open if and only if its intersection with each of the compact subsets is open, in the subspace topology. | A [[topological space]] is said to be '''compactly generated''' if the topology on it is generated by a collection of compact subsets. In other words, a set in the topological space is open if and only if its intersection with each of the compact subsets is open, in the subspace topology. | ||
===Definition with symbols=== | |||
A topological space <math>X</math> is said to be '''compactly generated''' if there exists a collection <math>\{ K_i \}_{i \in I}</math> of compact subsets of <math>X</math>, such that a subset <math>U \subset X</math> is open if and only if <math>U \cap K_i</math> is open in <math>K_i</math> for every <math>i \in I</math>. | |||
==Relation with other properties== | ==Relation with other properties== | ||
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===Stronger properties=== | ===Stronger properties=== | ||
{| class="soritable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[Weaker than::compact space]] || every open cover has a finite subcover || [[compact implies compactly generated]] || [[compactly generated not implies compact]] || {{intermediate notions short|compactly generated space|compact space}} | |||
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| [[Weaker than::locally compact space]] || every point is contained in an open subset contained in a closed, compact subset || [[locally compact implies compactly generated]] || [[compactly generated not implies locally compact]] || {{intermediate notions short|compactly generated space|locally compact space}} | |||
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| [[Weaker than::first-countable space]] || countable basis at every point ||[[first-countable implies compactly generated]] || [[compactly generated not implies first-countable]] || {{intermediate notions short|compactly generated space|first-countable space}} | |||
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| [[Weaker than::metrizable space]] || underlying topology of a [[metric space]] || [[metrizable implies compactly generated]] || [[compactly generated not implies metrizable]] || {{intermediate notions short|compactly generated space|metrizable space}} | |||
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| [[Weaker than::CW-space]] || underlying topology of a [[CW-complex]] || [[CW implies compactly generated]] || [[compactly generated not implies CW]] || {{intermediate notions short|compactly generated space|CW-space}} | |||
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| [[Weaker than::compactly generated Hausdorff space]] || compactly generated and [[Hausdorff space|Hausdorff]] || || || {{intermediate notions short|compactly generated space|compactly generated Hausdorff space}} | |||
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==Metaproperties== | ==Metaproperties== | ||
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==References== | ==References== | ||
===Textbook references=== | ===Textbook references=== | ||
* {{booklink|Munkres}}, Page 283 (formal definition) | * {{booklink-defined|Munkres}}, Page 283 (formal definition) | ||
Latest revision as of 20:43, 26 January 2012
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 compactly generated if the topology on it is generated by a collection of compact subsets. In other words, a set in the topological space is open if and only if its intersection with each of the compact subsets is open, in the subspace topology.
Definition with symbols
A topological space is said to be compactly generated if there exists a collection of compact subsets of , such that a subset is open if and only if is open in for every .
Relation with other properties
Stronger properties
Metaproperties
References
Textbook references
- Topology (2nd edition) by James R. MunkresMore info, Page 283 (formal definition)