Sober T1 space - Revision history

Vipul: /* Weaker properties */

2017-01-05T01:27:23Z

Weaker properties

← Older revision		Revision as of 01:27, 5 January 2017
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	\| [[Stronger than::sober space]] \|\| the only irreducible closed subsets are closures of one-point subsets \|\| \|\| \|\| {{intermediate notions short\|sober space\|sober T1 space}}		\| [[Stronger than::sober space]] \|\| the only irreducible closed subsets are closures of one-point subsets \|\| \|\| \|\| {{intermediate notions short\|sober space\|sober T1 space}}
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			\| [[Stronger than::sober T0 space]] \|\| sober and a [[Kolmogorov space]]: any two points are topologically distinguishable \|\| \|\| \|\| {{intermediate notions short\|sober T0 space\|sober T1 space}}
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	\| [[Stronger than::T1 space]] \|\| every singleton subset is closed \|\| \|\| \|\| {{intermediate notions short\|T1 space\|sober T1 space}}		\| [[Stronger than::T1 space]] \|\| every singleton subset is closed \|\| \|\| \|\| {{intermediate notions short\|T1 space\|sober T1 space}}
	\|}		\|}

Vipul: /* Definition */

2017-01-05T01:26:31Z

Definition

← Older revision		Revision as of 01:26, 5 January 2017
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	* <math>A</math> is a singleton subset, i.e., it has precisely one element.		* <math>A</math> is a singleton subset, i.e., it has precisely one element.
	* <math>A</math> is an [[irreducible closed subset]] of <math>X</math>, i.e., it is a [[closed subset]] and cannot be expressed as a union of two proper closed subsets of it (note that it does not matter for this definition whether we consider "closed" inside <math>A</math> via the [[subspace topology]], or closed inside <math>X</math>)		* <math>A</math> is an [[irreducible space\|irreducible]] [[closed subset]] of <math>X</math>, i.e., it is a [[closed subset]] and cannot be expressed as a union of two proper closed subsets of it (note that it does not matter for this definition whether we consider "closed" inside <math>A</math> via the [[subspace topology]], or closed inside <math>X</math>)

	== Relation with other properties ==		== Relation with other properties ==

Vipul: Created page with "{{topospace property conjunction|sober space|T1 space}} == Definition == A topological space $X$ is termed a '''sober T1 space''' if, for any nonempty subset..."

2017-01-05T01:24:35Z

Created page with "{{topospace property conjunction|sober space|T1 space}} == Definition == A topological space <math>X</math> is termed a '''sober T1 space''' if, for any nonempty subset..."

New page

{{topospace property conjunction|sober space|T1 space}}

== Definition ==

A [[topological space]] <math>X</math> is termed a '''sober T1 space''' if, for any nonempty subset <math>A</math> of <math>X</math>, the following are equivalent:

* <math>A</math> is a singleton subset, i.e., it has precisely one element.
* <math>A</math> is an [[irreducible closed subset]] of <math>X</math>, i.e., it is a [[closed subset]] and cannot be expressed as a union of two proper closed subsets of it (note that it does not matter for this definition whether we consider "closed" inside <math>A</math> via the [[subspace topology]], or closed inside <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::Hausdorff space]] || any two distinct points can be separated by disjoint open subsets || [[Hausdorff implies sober]] || || {{intermediate notions short|sober T1 space|Hausdorff space}}
|-
| [[Weaker than::compact sober T1 space]] || [[compact space]] as well as a sober T1 space || || || {{intermediate notions short|sober T1 space|compact sober T1 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 one-point subsets || || || {{intermediate notions short|sober space|sober T1 space}}
|-
| [[Stronger than::T1 space]] || every singleton subset is closed || || || {{intermediate notions short|T1 space|sober T1 space}}
|}

Sober T1 space - Revision history

Vipul: /* Weaker properties */

Vipul: /* Definition */

Vipul: Created page with "{{topospace property conjunction|sober space|T1 space}} == Definition == A topological space X is termed a '''sober T1 space''' if, for any nonempty subset..."

Vipul: Created page with "{{topospace property conjunction|sober space|T1 space}} == Definition == A topological space $X$ is termed a '''sober T1 space''' if, for any nonempty subset..."