Pseudocompact space: Difference between revisions
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==Definition== | ==Definition== | ||
A [[topological space]] is said to be '''pseudocompact''' if, | ===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== | ||
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===Stronger properties=== | ===Stronger properties=== | ||
{| class="wikitable" border="1" | |||
! 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]] | ||
Revision as of 00:25, 24 December 2009
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 is a variation of compactness. View other variations of compactness
Definition
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 information: Equivalence of definitions of pseudocompact space
Relation with other properties
Stronger properties
| property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions | |||||
|---|---|---|---|---|---|---|---|---|---|
| Compact space | Every open cover has a finite subcover | compact implies pseudocompact | pseudocompact not implies compact | Feebly compact space|FULL LIST, MORE INFO | 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 | |FULL LIST, MORE INFO |