Second-countable T1 space: Difference between revisions
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{{topospace property conjunction}} | {{topospace property conjunction|second-countable space|T1 space}} | ||
==Relation with other properties== | |||
===Stronger properties=== | |||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[Weaker than::Polish space]] || || || || | |||
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| [[Weaker than::separable metrizable space]] || || || || | |||
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===Weaker properties=== | |||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Stronger than::second-countable space]]|| || || || | |||
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| [[Stronger than::T1 space]] || || || || | |||
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| [[Stronger than::perfect space]] || || [[second-countable and T1 implies perfect]] || || | |||
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===Control of cardinality=== | |||
* [[Second-countable and T1 implies cardinality at most that of the continuum]] | |||
Latest revision as of 20:37, 26 January 2012
Definition
A topological space is termed a second-countable T1 space if it is both a second-countable space and a T1 space.
This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces
This article describes a property of topological spaces obtained as a conjunction of the following two properties: second-countable space and T1 space
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Polish space | ||||
| separable metrizable space |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| second-countable space | ||||
| T1 space | ||||
| perfect space | second-countable and T1 implies perfect |