Second-countable space: Difference between revisions
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==Definition== | ==Definition== | ||
A [[topological space]] is termed '''second-countable''' if it admits a countable [[basis of a | A [[topological space]] is termed '''second-countable''' if it satisfies the following equivalent conditions: | ||
* It admits a finite or countable [[basis]], i.e., a finite or countable collection of open subsets that form a basis for the topology. | |||
* It admits a finite countable [[subbasis]], i.e., a finite or countable collection of open subsets that form a subbasis for the topology. | |||
* The [[weight]] of the space is either finite or countable. | |||
==Relation with other properties== | ==Relation with other properties== | ||
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===Stronger properties=== | ===Stronger properties=== | ||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Weaker than::separable metrizable space]] || || || || | |||
|- | |||
| [[Weaker than::Polish space]] || || || || | |||
|- | |||
| [[Weaker than::Sub-Euclidean space]]|| || || || | |||
|- | |||
| [[Weaker than::second-countable T1 space]] || || || || | |||
|} | |||
===Weaker properties=== | ===Weaker properties=== | ||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Stronger than::hereditarily separable space]] || || || || | |||
|- | |||
| [[Stronger than::separable space]] || || [[second-countable implies separable]]||[[separable not implies second-countable]] || | |||
|- | |||
| [[Stronger than::first-countable space]] || ||[[second-countable implies first-countable]]||[[first-countable not implies second-countable]] || | |||
|- | |||
| [[Stronger than::Lindelof space]] || || [[second-countable implies Lindelof]]||[[Lindelof not implies second-countable]] || | |||
|- | |||
| [[Stronger than::compactly generated space]] || || [[first-countable implies compactly generated|via first-countable]] || [[compactly generated not implies first-countable|via first-countable]] || {{intermediate notions short|compactly generated space|second-countable space}} | |||
|} | |||
==Metaproperties== | |||
{{subspace-closed}} | |||
Any subspace of a second-countable space is second-countable. {{proofat|[[Second-countability is hereditary]]}} | |||
{{countable DP-closed}} | |||
==References== | |||
===Textbook references=== | |||
* {{booklink-defined|Munkres}}, Page 190, Chapter 4, Section 30 (formal definition) | |||
Latest revision as of 04:21, 27 January 2012
This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces
Definition
A topological space is termed second-countable if it satisfies the following equivalent conditions:
- It admits a finite or countable basis, i.e., a finite or countable collection of open subsets that form a basis for the topology.
- It admits a finite countable subbasis, i.e., a finite or countable collection of open subsets that form a subbasis for the topology.
- The weight of the space is either finite or countable.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| separable metrizable space | ||||
| Polish space | ||||
| Sub-Euclidean space | ||||
| second-countable T1 space |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| hereditarily separable space | ||||
| separable space | second-countable implies separable | separable not implies second-countable | ||
| first-countable space | second-countable implies first-countable | first-countable not implies second-countable | ||
| Lindelof space | second-countable implies Lindelof | Lindelof not implies second-countable | ||
| compactly generated space | via first-countable | via first-countable | |FULL LIST, MORE INFO |
Metaproperties
Hereditariness
This property of topological spaces is hereditary, or subspace-closed. In other words, any subspace (subset with the subspace topology) of a topological space with this property also has this property.
View other subspace-hereditary properties of topological spaces
Any subspace of a second-countable space is second-countable. For full proof, refer: Second-countability is hereditary
References
Textbook references
- Topology (2nd edition) by James R. MunkresMore info, Page 190, Chapter 4, Section 30 (formal definition)