Ultraconnected space - Revision history

Vipul: /* Opposite properties */

2017-01-05T00:45:48Z

Opposite properties

← Older revision		Revision as of 00:45, 5 January 2017
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	* [[T1 space]]: See [[ultraconnected and T1 implies one-point space]]		* [[T1 space]]: See [[ultraconnected and T1 implies one-point space]]
	* [[Compact space]]: See [[ultraconnected and compact implies one-point space]]

	===Similar properties===		===Similar properties===

	* [[Irreducible space]], with a similar definition but the roles of "open" and "closed" interchanged		* [[Irreducible space]], with a similar definition but the roles of "open" and "closed" interchanged

Vipul: /* Opposite properties */

2017-01-05T00:42:38Z

Opposite properties

← Older revision		Revision as of 00:42, 5 January 2017
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	* [[T1 space]]: See [[ultraconnected and T1 implies one-point space]]		* [[T1 space]]: See [[ultraconnected and T1 implies one-point space]]
			* [[Compact space]]: See [[ultraconnected and compact implies one-point space]]

	===Similar properties===		===Similar properties===

	* [[Irreducible space]], with a similar definition but the roles of "open" and "closed" interchanged		* [[Irreducible space]], with a similar definition but the roles of "open" and "closed" interchanged

Vipul: /* Definition */

2017-01-05T00:40:41Z

Definition

← Older revision		Revision as of 00:40, 5 January 2017
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	A [[topological space]] is termed an '''ultraconnected space''' if it satisfies the following equivalent conditions:		A [[topological space]] is termed an '''ultraconnected space''' if it satisfies the following equivalent conditions:

	# It cannot be expressed as a union of two proper open subsets		# It is nonempty and cannot be expressed as a union of two proper open subsets
	# It cannot be expressed as a union of finitely many proper open subsets		# It is nonempty and cannot be expressed as a union of finitely many proper open subsets
	# ~~Any~~ two nonempty closed subsets have nonempty intersection		# It is nonempty and any two nonempty closed subsets have nonempty intersection

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

Vipul: /* Relation with other properties */

2017-01-05T00:40:19Z

Relation with other properties

← Older revision		Revision as of 00:40, 5 January 2017
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	* [[T1 space]]: See [[ultraconnected and T1 implies one-point space]]		* [[T1 space]]: See [[ultraconnected and T1 implies one-point space]]

			===Similar properties===

			* [[Irreducible space]], with a similar definition but the roles of "open" and "closed" interchanged

Vipul: /* Definition */

2017-01-05T00:39:39Z

Definition

← Older revision		Revision as of 00:39, 5 January 2017
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	==Definition==		==Definition==

	A [[topological space]] is termed an '''ultraconnected space''' if it is a ~~non-empty space and any~~ two ~~non-empty~~ closed subsets have ~~non-empty~~ intersection.		A [[topological space]] is termed an '''ultraconnected space''' if it satisfies the following equivalent conditions:

			# It cannot be expressed as a union of two proper open subsets
			# It cannot be expressed as a union of finitely many proper open subsets
			# Any two nonempty closed subsets have nonempty intersection

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

Vipul: /* Definition */

2017-01-05T00:37:26Z

Definition

← Older revision		Revision as of 00:37, 5 January 2017
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	==Definition==		==Definition==

	A [[topological space]] is termed an '''ultraconnected space''' if it is a non-empty space and an two non-empty ~~disjoint~~ closed subsets have non-empty intersection.		A [[topological space]] is termed an '''ultraconnected space''' if it is a non-empty space and any two non-empty closed subsets have non-empty intersection.

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

Vipul: /* Relation with other properties */

2012-01-28T17:41:50Z

Relation with other properties

← Older revision		Revision as of 17:41, 28 January 2012
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	! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-
	\| [[Stronger than::path-connected space]] \|\| \|\| \|\| \|\|		\| [[Stronger than::path-connected space]] \|\| \|\| [[ultraconnected implies path-connected]] \|\| \|\| {{intermediate notions short\|path-connected space\|ultraconnected space}}
	\|-		\|-
	\| [[Stronger than::connected space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|connected space\|ultraconnected space}}		\| [[Stronger than::connected space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|connected space\|ultraconnected space}}
	\|-		\|-
	\| [[Stronger than::normal space]] \|\| \|\| \|\| \|\|		\| [[Stronger than::normal space]] \|\| \|\| [[ultraconnected implies normal]] \|\| \|\| {{intermediate notions short\|normal space\|ultraconnected space}}
	\|-		\|-
	\| [[Stronger than::pseudocompact space]] \|\| \|\| \|\| \|\|		\| [[Stronger than::pseudocompact space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|pseudocompact space\|ultraconnected space}}
	\|-		\|-
	\| [[Stronger than::limit point-compact space]] \|\| \|\| \|\| \|\|		\| [[Stronger than::limit point-compact space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|limit point-compact space\|ultraconnected space}}
	\|}		\|}

Vipul: /* Relation with other properties */

2012-01-28T17:40:25Z

Relation with other properties

← Older revision		Revision as of 17:40, 28 January 2012
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	\| [[Stronger than::connected space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|connected space\|ultraconnected space}}		\| [[Stronger than::connected space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|connected space\|ultraconnected space}}
	\|-		\|-
	\| [[Stronger than::normal~~-minus-Hausdorff~~ space]] \|\| \|\| \|\| \|\|		\| [[Stronger than::normal space]] \|\| \|\| \|\| \|\|
	\|-		\|-
	\| [[Stronger than::pseudocompact space]] \|\| \|\| \|\| \|\|		\| [[Stronger than::pseudocompact space]] \|\| \|\| \|\| \|\|

Vipul: /* Weaker properties */

2012-01-26T19:29:14Z

Weaker properties

← Older revision		Revision as of 19:29, 26 January 2012
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	\|-		\|-
	\| [[Stronger than::path-connected space]] \|\| \|\| \|\| \|\|		\| [[Stronger than::path-connected space]] \|\| \|\| \|\| \|\|
			\|-
			\| [[Stronger than::connected space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|connected space\|ultraconnected space}}
	\|-		\|-
	\| [[Stronger than::normal-minus-Hausdorff space]] \|\| \|\| \|\| \|\|		\| [[Stronger than::normal-minus-Hausdorff space]] \|\| \|\| \|\| \|\|

Vipul: Created page with "==Definition== A topological space is termed an '''ultraconnected space''' if it is a non-empty space and an two non-empty disjoint closed subsets have non-empty intersec..."

2012-01-26T19:28:11Z

Created page with "==Definition== A topological space is termed an '''ultraconnected space''' if it is a non-empty space and an two non-empty disjoint closed subsets have non-empty intersec..."

New page

==Definition==

A [[topological space]] is termed an '''ultraconnected space''' if it is a non-empty space and an two non-empty disjoint closed subsets have non-empty intersection.

==Relation with other properties==

===Weaker properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::path-connected space]] || || || ||
|-
| [[Stronger than::normal-minus-Hausdorff space]] || || || ||
|-
| [[Stronger than::pseudocompact space]] || || || ||
|-
| [[Stronger than::limit point-compact space]] || || || ||
|}

===Opposite properties===

* [[T1 space]]: See [[ultraconnected and T1 implies one-point space]]