Second-countable space: Difference between revisions
m (7 revisions) |
No edit summary |
||
| Line 9: | Line 9: | ||
===Stronger properties=== | ===Stronger properties=== | ||
* [[Separable metrizable space]] | * [[Weaker than::Separable metrizable space]] | ||
* [[Polish space]] | * [[Weaker than::Polish space]] | ||
* [[Sub-Euclidean space]] | * [[Weaker than::Sub-Euclidean space]] | ||
===Weaker properties=== | ===Weaker properties=== | ||
* [[Hereditarily separable space]] | * [[Stronger than::Hereditarily separable space]] | ||
* [[Separable space]] | * [[Stronger than::Separable space]] | ||
* [[First-countable space]] | * [[Stronger than::First-countable space]] | ||
* [[Stronger than::Lindelof space]] | |||
==Metaproperties== | ==Metaproperties== | ||
| Line 23: | Line 24: | ||
{{subspace-closed}} | {{subspace-closed}} | ||
Any subspace of a second-countable space is second-countable. | Any subspace of a second-countable space is second-countable. {{proofat|[[Second-countability is hereditary]]}} | ||
{{countable DP-closed}} | {{countable DP-closed}} | ||
==References== | ==References== | ||
===Textbook references=== | ===Textbook references=== | ||
* {{booklink|Munkres}}, Page 190 (formal definition) | * {{booklink-defined|Munkres}}, Page 190, Chapter 4, Section 30 (formal definition) | ||
Revision as of 21:04, 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
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)