Second-countable space

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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

Template:Countable DP-closed

References

Textbook references

  • Topology (2nd edition) by James R. MunkresMore info, Page 190, Chapter 4, Section 30 (formal definition)