Editing Compactly generated space (section)

==Examples==

As explained in the [[#Definitions|Definitions]] section, there is no universally accepted definition in the literature for compactly generated spaces; but Definitions 1, 2, 3 from that section are some of the more commonly used.  In order to express results in a more concise way, this section will make use of the abbreviations '''CG-1''', '''CG-2''', '''CG-3''' to denote each of the three definitions unambiguously.  This is summarized in the table below (see the Definitions section for other equivalent conditions for each).

{| class="wikitable"
|-
! Abbreviation !! Meaning summary
|-
| [[#Definition 1|CG-1]]
| Topology coherent with family of its compact subspaces
|-
| [[#Definition 2|CG-2]]
| Topology same as final topology with respect to continuous maps from arbitrary compact Hausdorff spaces 
|-
| [[#Definition 3|CG-3]]
| Topology coherent with family of its compact Hausdorff subspaces
|}

For Hausdorff spaces the properties CG-1, CG-2, CG-3 are equivalent.  Such spaces can be called ''compactly generated Hausdorff'' without ambiguity.

Every CG-3 space is CG-2 and every CG-2 space is CG-1.  The converse implications do not hold in general, as shown by some of the examples below.

For [[weak Hausdorff]] spaces the properties CG-2 and CG-3 are equivalent.{{sfn|Strickland|2009|loc=Lemma 1.4(c)}}

[[Sequential space]]s are CG-2.{{sfn|Strickland|2009|loc=Proposition 1.6}}  This includes [[first countable space]]s, [[Alexandrov-discrete space]]s, [[finite space]]s.

Every CG-3 space is a [[T1 space|T<sub>1</sub> space]] (because given a singleton <math>\{x\}\subseteq X,</math> its intersection with every compact Hausdorff subspace <math>K\subseteq X</math> is the empty set or a single point, which is closed in <math>K;</math> hence the singleton is closed in <math>X</math>).  Finite T<sub>1</sub> spaces have the [[discrete topology]].  So among the finite spaces, which are all CG-2, the CG-3 spaces are the ones with the discrete topology.  Any finite non-discrete space, like the [[Sierpiński space]], is an example of CG-2 space that is not CG-3.

[[Compact space]]s and [[weakly locally compact]] spaces are CG-1, but not necessarily CG-2 (see examples below).

Compactly generated Hausdorff spaces include the Hausdorff version of the various classes of spaces mentioned above as CG-1 or CG-2, namely Hausdorff sequential spaces, Hausdorff first countable spaces, [[locally compact Hausdorff]] spaces, etc.  In particular, [[metric space]]s and [[topological manifold]]s are compactly generated.  [[CW complex]]es are also Hausdorff compactly generated.

To provide examples of spaces that are not compactly generated, it is useful to examine ''anticompact''<ref>{{cite journal |last1=Bankston |first1=Paul |title=The total negation of a topological property |journal=Illinois Journal of Mathematics |date=1979 |volume=23 |issue=2 |pages=241–252 |doi=10.1215/ijm/1256048236|doi-access=free }}</ref> spaces, that is, spaces whose compact subspaces are all finite.  If a space <math>X</math> is anticompact and T<sub>1</sub>, every compact subspace of <math>X</math> has the discrete topology and the corresponding k-ification of <math>X</math> is the discrete topology.  Therefore, any anticompact T<sub>1</sub> non-discrete space is not CG-1.  Examples include:
* The [[cocountable topology]] on an uncountable space.
* The one-point Lindelöfication of an uncountable discrete space (also called [[Fortissimo space]]).
* The [[Arens-Fort space]].
* The [[Appert space]].
* The "Single ultrafilter topology".{{sfn|Steen|Seebach|1995|loc=Example 114, p. 136}}

Other examples of (Hausdorff) spaces that are not compactly generated include:
* The [[product (topology)|product]] of uncountably many copies of <math>\mathbb R</math> (each with the usual [[Euclidean topology]]).{{sfn|Willard|2004|loc=Problem 43H(2)}}
* The product of uncountably many copies of <math>\mathbb Z</math> (each with the [[discrete topology]]).

For examples of spaces that are CG-1 and not CG-2, one can start with any space <math>Y</math> that is not CG-1 (for example the Arens-Fort space or an uncountable product of copies of <math>\mathbb R</math>) and let <math>X</math> be the [[one-point compactification]] of <math>Y.</math>  The space <math>X</math> is compact, hence CG-1.  But it is not CG-2 because open subspaces inherit the CG-2 property and <math>Y</math> is an open subspace of <math>X</math> that is not CG-2.