Pseudocompact space
From Topospaces
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
Contents
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. 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 spaces.
Equivalence of definitions
Further information: Equivalence of definitions of pseudocompact space
Relation with other properties
Stronger properties
| Property | Meaning | 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 |
