Pseudocompact space - Revision history

Vipul: /* Relation with other properties */

2010-10-20T16:20:52Z

Relation with other properties

← Older revision		Revision as of 16:20, 20 October 2010
Line 24:		Line 24:
	! 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
	\|-		\|-
	\| [[Weaker than::~~Compact~~ space]] \|\| Every open cover has a finite subcover \|\| [[compact implies pseudocompact]] \|\| [[pseudocompact not implies compact]] \|\| {{intermediate notions short\|pseudocompact space\|compact space}}		\| [[Weaker than::compact space]] \|\| Every open cover has a finite subcover \|\| [[compact implies pseudocompact]] \|\| [[pseudocompact not implies compact]] \|\| {{intermediate notions short\|pseudocompact space\|compact space}}
	\|-		\|-
	\| [[Weaker than::~~Feebly~~ compact space]] (also called lightly compact space) \|\|Every locally finite collection of nonempty open subsets is finite \|\| [[feebly compact implies pseudocompact]] \|\| [[pseudocompact not implies feebly compact]] \|\| {{intermediate notions short\|pseduocompact space\|feebly compact space}}		\| [[Weaker than::feebly compact space]] (also called lightly compact space) \|\|Every locally finite collection of nonempty open subsets is finite \|\| [[feebly compact implies pseudocompact]] \|\| [[pseudocompact not implies feebly compact]] \|\| {{intermediate notions short\|pseduocompact space\|feebly compact space}}
	\|}		\|}

Vipul at 16:20, 20 October 2010

2010-10-20T16:20:33Z

← Older revision		Revision as of 16:20, 20 October 2010
Line 9:		Line 9:
	A [[topological space]] is said to be '''pseudocompact''' if it satisfies the following equivalent properties:		A [[topological space]] is said to be '''pseudocompact''' if it satisfies the following equivalent properties:

	# For any continuous map from the topological space to the [[real line]], the image of the topological space is a closed and bounded subset of the real line.		# For any [[continuous map]] from the topological space to the [[real line]], the image of the topological space is a closed and bounded subset of the real line.
	# For any [[continuous map]] from the topological space to the [[real line]], the image of the topological space is a bounded subset of the real line.		# For any [[continuous map]] from the topological space to the [[real line]], the image of the topological space is a bounded subset of the real line.
	# Any continuous map from the topological space to the real line attains its absolute maximum and its absolute minimum. Note that this final formulation is equivalent to saying that the space satisfies the conclusion of the [[extreme value theorem]], which is stated for [[compact space]]s.		# Any continuous map from the topological space to the real line attains its absolute maximum and its absolute minimum. Note that this final formulation is equivalent to saying that the space satisfies the conclusion of the [[extreme value theorem]], which is stated for [[compact space]]s.
Line 21:		Line 21:
	===Stronger properties===		===Stronger properties===

	{\| class="~~wikitable~~" border="1"		{\| class="sortable" border="1"
	! ~~property~~ !! ~~quick description~~ !! ~~proof~~ of implication !! ~~proof~~ of strictness (reverse implication failure) !! ~~intermediate~~ notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-
	\| [[Weaker than::Compact space]] \|\| Every open cover has a finite subcover \|\| [[compact implies pseudocompact]] \|\| [[pseudocompact not implies compact]] \|\| {{intermediate notions short\|pseudocompact space\|compact space}}		\| [[Weaker than::Compact space]] \|\| Every open cover has a finite subcover \|\| [[compact implies pseudocompact]] \|\| [[pseudocompact not implies compact]] \|\| {{intermediate notions short\|pseudocompact space\|compact space}}

Vipul: /* Definition */

2009-12-24T00:58:18Z

Definition

← Older revision		Revision as of 00:58, 24 December 2009
Line 11:		Line 11:
	# For any continuous map from the topological space to the [[real line]], the image of the topological space is a closed and bounded subset of the real line.		# For any continuous map from the topological space to the [[real line]], the image of the topological space is a closed and bounded subset of the real line.
	# For any [[continuous map]] from the topological space to the [[real line]], the image of the topological space is a bounded subset of the real line.		# For any [[continuous map]] from the topological space to the [[real line]], the image of the topological space is a bounded subset of the real line.
	# Any continuous map from the topological space to the real line attains its absolute maximum and its absolute minimum.		# Any continuous map from the topological space to the real line attains its absolute maximum and its absolute minimum. Note that this final formulation is equivalent to saying that the space satisfies the conclusion of the [[extreme value theorem]], which is stated for [[compact space]]s.

	===Equivalence of definitions===		===Equivalence of definitions===

Vipul: /* Relation with other properties */

2009-12-24T00:30:13Z

Relation with other properties

← Older revision		Revision as of 00:30, 24 December 2009
Line 25:		Line 25:
	\|-		\|-
	\| [[Weaker than::Compact space]] \|\| Every open cover has a finite subcover \|\| [[compact implies pseudocompact]] \|\| [[pseudocompact not implies compact]] \|\| {{intermediate notions short\|pseudocompact space\|compact space}}		\| [[Weaker than::Compact space]] \|\| Every open cover has a finite subcover \|\| [[compact implies pseudocompact]] \|\| [[pseudocompact not implies compact]] \|\| {{intermediate notions short\|pseudocompact space\|compact space}}
			\|-
	\| [[Weaker than::Feebly compact space]] (also called lightly compact space) \|\|Every locally finite collection of nonempty open subsets is finite \|\| [[feebly compact implies pseudocompact]] \|\| [[pseudocompact not implies feebly compact]] \|\| {{intermediate notions short\|pseduocompact space\|feebly compact space}}		\| [[Weaker than::Feebly compact space]] (also called lightly compact space) \|\|Every locally finite collection of nonempty open subsets is finite \|\| [[feebly compact implies pseudocompact]] \|\| [[pseudocompact not implies feebly compact]] \|\| {{intermediate notions short\|pseduocompact space\|feebly compact space}}
	\|}		\|}

Vipul: /* Symbol-free definition */

2009-12-24T00:29:51Z

Symbol-free definition

← Older revision		Revision as of 00:29, 24 December 2009
Line 9:		Line 9:
	A [[topological space]] is said to be '''pseudocompact''' if it satisfies the following equivalent properties:		A [[topological space]] is said to be '''pseudocompact''' if it satisfies the following equivalent properties:

			# For any continuous map from the topological space to the [[real line]], the image of the topological space is a closed and bounded subset of the real line.
	# For any [[continuous map]] from the topological space to the [[real line]], the image of the topological space is a bounded subset of the real line.		# For any [[continuous map]] from the topological space to the [[real line]], the image of the topological space is a bounded subset of the real line.
	~~# For any continuous map from the topological space to the [[real line]], the image of the topological space is a closed and bounded subset of the real line.~~
	# Any continuous map from the topological space to the real line attains its absolute maximum and its absolute minimum.		# Any continuous map from the topological space to the real line attains its absolute maximum and its absolute minimum.

Vipul at 00:25, 24 December 2009

2009-12-24T00:25:37Z

← Older revision		Revision as of 00:25, 24 December 2009
Line 5:		Line 5:
	==Definition==		==Definition==

	A [[topological space]] is said to be '''pseudocompact''' if, ~~under~~ any continuous map to the real line, ~~its~~ image is bounded.		===Symbol-free definition===

			A [[topological space]] is said to be '''pseudocompact''' if it satisfies the following equivalent properties:

			# For any [[continuous map]] from the topological space to the [[real line]], the image of the topological space is a bounded subset of the real line.
			# For any continuous map from the topological space to the [[real line]], the image of the topological space is a closed and bounded subset of the real line.
			# Any continuous map from the topological space to the real line attains its absolute maximum and its absolute minimum.

			===Equivalence of definitions===

			{{further\|[[Equivalence of definitions of pseudocompact space]]}}

	==Relation with other properties==		==Relation with other properties==
Line 11:		Line 21:
	===Stronger properties===		===Stronger properties===

	* [[Compact space]]		{\| class="wikitable" border="1"
	* [[Feebly compact space]] (also called lightly compact space)		! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
			\|-
			\| [[Weaker than::Compact space]] \|\| Every open cover has a finite subcover \|\| [[compact implies pseudocompact]] \|\| [[pseudocompact not implies compact]] \|\| {{intermediate notions short\|pseudocompact space\|compact space}}
			\| [[Weaker than::Feebly compact space]] (also called lightly compact space) \|\|Every locally finite collection of nonempty open subsets is finite \|\| [[feebly compact implies pseudocompact]] \|\| [[pseudocompact not implies feebly compact]] \|\| {{intermediate notions short\|pseduocompact space\|feebly compact space}}
			\|}

	===Related properties===		===Related properties===

	* [[Realcompact space]]		* [[Realcompact space]]

Vipul: 3 revisions

2008-05-11T19:57:41Z

3 revisions

← Older revision	Revision as of 19:57, 11 May 2008
(No difference)

Vipul: /* Stronger properties */

2007-08-18T13:22:34Z

Stronger properties

← Older revision		Revision as of 13:22, 18 August 2007
Line 12:		Line 12:

	* [[Compact space]]		* [[Compact space]]
			* [[Feebly compact space]] (also called lightly compact space)

	===Related properties===		===Related properties===

	* [[Realcompact space]]		* [[Realcompact space]]

Vipul: /* Definition */

2007-08-18T08:48:21Z

Definition

← Older revision		Revision as of 08:48, 18 August 2007
Line 5:		Line 5:
	==Definition==		==Definition==

	A [[topological space]] is said to be '''pseudocompact''' if, under any continuous map to the real line, its image is ~~compact~~.		A [[topological space]] is said to be '''pseudocompact''' if, under any continuous map to the real line, its image is bounded.

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

Vipul at 08:09, 18 August 2007

2007-08-18T08:09:41Z

New page

{{topospace property}}

{{variationof|compactness}}

==Definition==

A [[topological space]] is said to be '''pseudocompact''' if, under any continuous map to the real line, its image is compact.

==Relation with other properties==

===Stronger properties===

* [[Compact space]]

===Related properties===

* [[Realcompact space]]