Second-countable space: Difference between revisions
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* [[Stronger than::Hereditarily separable space]] | * [[Stronger than::Hereditarily separable space]] | ||
* [[Stronger than::Separable space]] | * [[Stronger than::Separable space]]: {{proofofstrictimplicationat|[[Second-countable implies separable]]|[[Separable not implies second-countable]]}} | ||
* [[Stronger than::First-countable space]] | * [[Stronger than::First-countable space]]: {{proofofstrictimplicationat|[[Second-countable implies first-countable]]|[[First-countable not implies second-countable]]}} | ||
* [[Stronger than::Lindelof space]] | * [[Stronger than::Lindelof space]]: {{proofofstrictimplicationat|[[Second-countable implies Lindelof]]|[[Lindelof not implies second-countable]]}} | ||
==Metaproperties== | ==Metaproperties== | ||
Revision as of 21:26, 20 July 2008
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 admits a countable basis.
Relation with other properties
Stronger properties
Weaker properties
- Hereditarily separable space
- Separable space: For proof of the implication, refer Second-countable implies separable and for proof of its strictness (i.e. the reverse implication being false) refer Separable not implies second-countable
- First-countable space: For proof of the implication, refer Second-countable implies first-countable and for proof of its strictness (i.e. the reverse implication being false) refer First-countable not implies second-countable
- Lindelof space: For proof of the implication, refer Second-countable implies Lindelof and for proof of its strictness (i.e. the reverse implication being false) refer Lindelof not implies second-countable
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)