KC-space: Difference between revisions

Latest revision as of 23:23, 17 July 2013

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 Hausdorffness. View other variations of Hausdorffness

Definition

Symbol-free definition

A topological space is termed a KC-space if every compact subset of it is closed (here, by compact subset, we mean a subset which is a compact space under the subspace topology).

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Hausdorff space	any two distinct points can be separated by disjoint open subsets	Hausdorff implies KC	KC not implies Hausdorff	Weakly Hausdorff space\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
weakly Hausdorff space	any subset of it arising as the continuous image of a compact Hausdorff space is closed in it			\|FULL LIST, MORE INFO
US-space		KC implies US	US not implies KC	\|FULL LIST, MORE INFO
T1 space	points are closed	KC implies T1	T1 not implies KC	\|FULL LIST, MORE INFO
Kolmogorov space	any two points can be distinguished	(via T1)	(via T1)	\|FULL LIST, MORE INFO

@@ Line 12: / Line 12: @@
 ===Stronger properties===
-* [[Hausdorff space]]: {{prpofofstrictimplicationat|[[Hausdorff implies KC]]|[[KC not implies Hausdorff]]}}
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::Hausdorff space]] || any two distinct points can be separated by disjoint open subsets|| [[Hausdorff implies KC]]|| [[KC not implies Hausdorff]] || {{intermediate notions short|KC-space|Hausdorff space}}
+|}
 ===Weaker properties===
-* [[US-space]]:{{proofat|[[KC implies US]]|[[US not implies KC]]}}
+{| class="sortable" border="1"
-* [[T1 space]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::weakly Hausdorff space]] || any subset of it arising as the continuous image of a [[compact Hausdorff space]] is closed in it || || || {{intermediate notions short|weakly Hausdorff space|KC-space}}
+|-
+| [[Stronger than::US-space]] || || [[KC implies US]]||[[US not implies KC]] || {{intermediate notions short|US-space|KC-space}}
+|-
+| [[Stronger than::T1 space]] || points are closed || [[KC implies T1]] || [[T1 not implies KC]] ||{{intermediate notions short|T1 space|KC-space}}
+|-
+| [[Stronger than::Kolmogorov space]] || any two points can be distinguished || (via T1) || (via T1) || {{intermediate notions short|Kolmogorov space|KC-space}}
+|}