Second-countable space: Difference between revisions

Revision as of 20:41, 26 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 countable basis, i.e., a 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.

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	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. Munkres^{More info}, Page 190, Chapter 4, Section 30 (formal definition)

@@ Line 12: / Line 12: @@
 ===Stronger properties===
-* [[Weaker than::Separable metrizable space]]
+{| class="sortable" border="1"
-* [[Weaker than::Polish space]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Weaker than::Sub-Euclidean space]]
+|-
+| [[Weaker than::separable metrizable space]] || || || ||
+|-
+| [[Weaker than::Polish space]] || || || ||
+|-
+| [[Weaker than::Sub-Euclidean space]]|| || || ||
+|-
+| [[Weaker than::second-countable T1 space]] || || || ||
+|}
 ===Weaker properties===
-* [[Stronger than::Hereditarily separable space]]
-* [[Stronger than::Separable space]]: {{proofofstrictimplicationat|[[Second-countable implies separable]]|[[Separable not implies second-countable]]}}
+{| class="sortable" border="1"
-* [[Stronger than::First-countable space]]: {{proofofstrictimplicationat|[[Second-countable implies first-countable]]|[[First-countable not implies second-countable]]}}
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Stronger than::Lindelof space]]: {{proofofstrictimplicationat|[[Second-countable implies Lindelof]]|[[Lindelof not implies second-countable]]}}
+|-
+| [[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==