Compactly generated space: Difference between revisions

Latest revision as of 20:43, 26 January 2012

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 compactly generated if the topology on it is generated by a collection of compact subsets. In other words, a set in the topological space is open if and only if its intersection with each of the compact subsets is open, in the subspace topology.

Definition with symbols

A topological space $X$ is said to be compactly generated if there exists a collection ${K_{i}}_{i \in I}$ of compact subsets of $X$ , such that a subset $U \subset X$ is open if and only if $U \cap K_{i}$ is open in $K_{i}$ for every $i \in I$ .

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 compactly generated	compactly generated not implies compact	\|FULL LIST, MORE INFO
locally compact space	every point is contained in an open subset contained in a closed, compact subset	locally compact implies compactly generated	compactly generated not implies locally compact	\|FULL LIST, MORE INFO
first-countable space	countable basis at every point	first-countable implies compactly generated	compactly generated not implies first-countable	\|FULL LIST, MORE INFO
metrizable space	underlying topology of a metric space	metrizable implies compactly generated	compactly generated not implies metrizable	\|FULL LIST, MORE INFO
CW-space	underlying topology of a CW-complex	CW implies compactly generated	compactly generated not implies CW	\|FULL LIST, MORE INFO
compactly generated Hausdorff space	compactly generated and Hausdorff			\|FULL LIST, MORE INFO

Metaproperties

References

Textbook references

Topology (2nd edition) by James R. Munkres^{More info}, Page 283 (formal definition)

@@ Line 17: / Line 17: @@
 ===Stronger properties===
-{| class="wikitable" border="1"
+{| class="soritable" border="1"
-! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 |-
-| [[Weaker than::Compact space]] || every open cover has a finite subcover || [[compact implies compactly generated]] || [[compactly generated not implies compact]] || {{intermediate notions short|compactly generated space|compact space}}
+| [[Weaker than::compact space]] || every open cover has a finite subcover || [[compact implies compactly generated]] || [[compactly generated not implies compact]] || {{intermediate notions short|compactly generated space|compact space}}
 |-
-| [[Weaker than::Locally compact space]] || every point is contained in an open subset contained in a closed, compact subset || [[locally compact implies compactly generated]] || [[compactly generated not implies locally compact]] || {{intermediate notions short|compactly generated space|locally compact space}}
+| [[Weaker than::locally compact space]] || every point is contained in an open subset contained in a closed, compact subset || [[locally compact implies compactly generated]] || [[compactly generated not implies locally compact]] || {{intermediate notions short|compactly generated space|locally compact space}}
 |-
-| [[Weaker than::First-countable space]] || countable basis at every point ||[[first-countable implies compactly generated]] || [[compactly generated not implies first-countable]] || {{intermediate notions short|compactly generated space|first-countable space}}
+| [[Weaker than::first-countable space]] || countable basis at every point ||[[first-countable implies compactly generated]] || [[compactly generated not implies first-countable]] || {{intermediate notions short|compactly generated space|first-countable space}}
 |-
-| [[Weaker than::Metrizable space]] || underlying topology of a [[metric space]] || [[metrizable implies compactly generated]] || [[compactly generated not implies metrizable]] || {{intermediate notions short|compactly generated space|metrizable space}}
+| [[Weaker than::metrizable space]] || underlying topology of a [[metric space]] || [[metrizable implies compactly generated]] || [[compactly generated not implies metrizable]] || {{intermediate notions short|compactly generated space|metrizable space}}
 |-
 | [[Weaker than::CW-space]] || underlying topology of a [[CW-complex]] || [[CW implies compactly generated]] || [[compactly generated not implies CW]] || {{intermediate notions short|compactly generated space|CW-space}}
 |-
-| [[Weaker than::Compactly generated Hausdorff space]] || compactly generated and [[Hausdorff space|Hausdorff]] || || || {{intermediate notions short|compactly generated space|compactly generated Hausdorff space}}
+| [[Weaker than::compactly generated Hausdorff space]] || compactly generated and [[Hausdorff space|Hausdorff]] || || || {{intermediate notions short|compactly generated space|compactly generated Hausdorff space}}
 |}