Vipul: /* = Weaker properties */

2017-01-05T01:31:52Z

= Weaker properties

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	=== Weaker properties ==		=== Weaker properties ===

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Vipul: Created page with "{{topospace property conjunction|sober space|T0 space}} == Statement == A topological space $X$ is termed a '''sober T0 space''' if it is both a sober space..."

2017-01-05T01:31:45Z

Created page with "{{topospace property conjunction|sober space|T0 space}} == Statement == A topological space <math>X</math> is termed a '''sober T0 space''' if it is both a sober space..."

New page

{{topospace property conjunction|sober space|T0 space}}

== Statement ==

A [[topological space]] <math>X</math> is termed a '''sober T0 space''' if it is both a [[sober space]] and a [[Kolmogorov space]] (T0 space). Explicitly:

* It is sober: the only irreducible closed subsets of <math>X</math> are closures of singleton subsets.
* It is T0: for any two distinct points <math>x,y \in X</math>, we can find either an open subset containing <math>x</math> and not <math>y</math>, or an open subset containing <math>y</math> and not <math>x</math>.

== Relation with other properties ==

=== Stronger properties ===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::sober T1 space]] || sober and every singleton subset is closed || || || {{intermediate notions short|sober T0 space|sober T1 space}}
|-
| [[Weaker than::Hausdorff space]] || any two distinct points can be separated by disjoint open subsets || || || {{intermediate notions short|sober T0 space|Hausdorff space}}
|}

=== Weaker properties ==

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::sober space]] || the only irreducible closed subsets are closures of singleton subsets || || || {{intermediate notions short|sober space|sober T0 space}}
|-
| [[Stronger than::Kolmogorov space]] || any two points are topologically distinguishable || || || {{intermediate notions short|Kolmogorov space|sober T0 space}}
|}

Sober T0 space - Revision history

Vipul: /* = Weaker properties */

Vipul: Created page with "{{topospace property conjunction|sober space|T0 space}} == Statement == A topological space X is termed a '''sober T0 space''' if it is both a sober space..."

Vipul: Created page with "{{topospace property conjunction|sober space|T0 space}} == Statement == A topological space $X$ is termed a '''sober T0 space''' if it is both a sober space..."