Second-countable space: Difference between revisions
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A [[topological space]] is termed '''second-countable''' if it satisfies the following equivalent conditions: | A [[topological space]] is termed '''second-countable''' if it satisfies the following equivalent conditions: | ||
* It admits a countable [[basis]], i.e., a countable collection of open subsets that form a basis for the topology. | * 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 countable [[subbasis]], i.e., a countable collection of open subsets that form a subbasis 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== | ||
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)