Feebly compact space: Difference between revisions
No edit summary |
|||
| (2 intermediate revisions by the same user not shown) | |||
| Line 5: | Line 5: | ||
==Definition== | ==Definition== | ||
A [[topological space]] is termed '''feebly compact''' if every [[locally finite collection]] of nonempty [[open subset]]s is finite. | A [[topological space]] is termed '''feebly compact''' or '''lightly compact''' if every [[locally finite collection]] of nonempty [[open subset]]s is finite. | ||
==Relation with other properties== | ==Relation with other properties== | ||
| Line 11: | Line 11: | ||
===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 feebly compact]] || [[feebly compact not implies compact]] || {{intermediate notions short|feebly compact space|compact space}} | |||
|} | |||
===Weaker properties=== | |||
{| class="wikitable" border="1" | |||
! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions | |||
|- | |||
| [[Stronger than::Pseudocompact space]] || Image of continuous map to reals is bounded || [[feebly compact implies pseudocompact]] || [[pseudocompact not implies feebly compact]] || {{intermediate notions short|pseudocompact space|feebly compact space}} | |||
|} | |||
Latest 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
A topological space is termed feebly compact or lightly compact if every locally finite collection of nonempty open subsets is finite.
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 feebly compact | feebly compact not implies compact | |FULL LIST, MORE INFO |
Weaker properties
| property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
|---|---|---|---|---|
| Pseudocompact space | Image of continuous map to reals is bounded | feebly compact implies pseudocompact | pseudocompact not implies feebly compact | |FULL LIST, MORE INFO |