Pseudocompact space: Difference between revisions
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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. | ||
# 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. | ||
Revision as of 00:29, 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 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.
- 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 |