Editing Compactly generated space (section)

==Properties==

(See the [[#Examples|Examples]] section for the meaning of the abbreviations CG-1, CG-2, CG-3.)

===Subspaces===

Subspaces of a compactly generated space are not compactly generated in general, even in the Hausdorff case.  For example, the [[ordinal space]] <math>\omega_1+1=[0,\omega_1]</math> where <math>\omega_1</math> is the [[first uncountable ordinal]] is compact Hausdorff, hence compactly generated.  Its subspace with all limit ordinals except <math>\omega_1</math> removed is isomorphic to the [[Fortissimo space]], which is not compactly generated (as  mentioned in the Examples section, it is anticompact and non-discrete).{{sfn|Lamartin|1977|p=8}}  Another example is the Arens space,{{sfn|Engelking|1989|loc=Example 1.6.19}}<ref>{{cite web |last=Ma |first=Dan |date=19 August 2010 |title=A note about the Arens' space |url=http://dantopology.wordpress.com/2010/08/18/a-note-about-the-arens-space/ }}</ref> which is sequential Hausdorff, hence compactly generated.  It contains as a subspace the [[Arens-Fort space]], which is not compactly generated.

In a CG-1 space, every closed set is CG-1.  The same does not hold for open sets.  For instance, as shown in the Examples section, there are many spaces that are not CG-1, but they are open in their [[one-point compactification]], which is CG-1.

In a CG-2 space <math>X,</math> every closed set is CG-2; and so is every open set (because there is a quotient map <math>q:Y\to X</math> for some locally compact Hausdorff space <math>Y</math> and for an open set <math>U\subseteq X</math> the restriction of <math>q</math> to <math>q^{-1}(U)</math> is also a quotient map on a locally compact Hausdorff space).  The same is true more generally for every [[locally closed]] set, that is, the intersection of an open set and a closed set.{{sfn|Lamartin|1977|loc=Proposition 1.8}}

In a CG-3 space, every closed set is CG-3.

===Quotients===

The [[disjoint union (topology)|disjoint union]] <math>{\coprod}_i X_i</math> of a family <math>(X_i)_{i\in I}</math> of topological spaces is CG-1 if and only if each space <math>X_i</math> is CG-1.  The corresponding statements also hold for CG-2{{sfn|Strickland|2009|loc=Proposition 2.2}}{{sfn|Rezk|2018|loc=Proposition 3.4(3)}} and CG-3.

A [[quotient space (topology)|quotient space]] of a CG-1 space is CG-1.{{sfn|Lawson|Madison|1974|p=3}}  In particular, every quotient space of a [[weakly locally compact]] space is CG-1.  Conversely, every CG-1 space <math>X</math> is the quotient space of a weakly locally compact space, which can be taken as the [[disjoint union (topology)|disjoint union]] of the compact subspaces of <math>X.</math>{{sfn|Lawson|Madison|1974|p=3}}

A quotient space of a CG-2 space is CG-2.{{sfn|Brown|2006|loc=5.9.1 (Corollary 2)}}  In particular, every quotient space of a [[locally compact Hausdorff]] space is CG-2.  Conversely, every CG-2 space is the quotient space of a locally compact Hausdorff space.{{sfn|Brown|2006|loc=Proposition 5.9.1}}{{sfn|Lamartin|1977|loc=Proposition 1.7}}

A quotient space of a CG-3 space is not CG-3 in general.  In fact, every CG-2 space is a quotient space of a CG-3 space (namely, some locally compact Hausdorff space); but there are CG-2 spaces that are not CG-3.  For a concrete example, the [[Sierpiński space]] is not CG-3, but is homeomorphic to the quotient of the compact interval <math>[0,1]</math> obtained by identifying <math>(0,1]</math> to a point.

More generally, any [[final topology]] on a set induced by a family of functions from CG-1 spaces is also CG-1.  And the same holds for CG-2.  This follows by combining the results above for disjoint unions and quotient spaces, together with the behavior of final topologies under composition of functions.

A [[wedge sum]] of CG-1 spaces is CG-1.  The same holds for CG-2.  This is also an application of the results above for disjoint unions and quotient spaces.

===Products===

The [[product (topology)|product]] of two compactly generated spaces need not be compactly generated, even if both spaces are Hausdorff and [[sequential space|sequential]].  For example, the space <math>X=\Reals \setminus \{1, 1/2, 1/3, \ldots\}</math> with the [[subspace topology]] from the real line is [[first countable]]; the space <math>Y=\Reals / \{1,2,3,\ldots\}</math> with the [[quotient topology]] from the real line with the positive integers identified to a point is sequential.  Both spaces are compactly generated Hausdorff, but their product <math>X\times Y</math> is not compactly generated.{{sfn|Engelking|1989|loc=Example 3.3.29}}

However, in some cases the product of two compactly generated spaces is compactly generated:
* The product of two first countable spaces is first countable, hence CG-2.
* The product of a CG-1 space and a [[locally compact]] space is CG-1.{{sfn|Lawson|Madison|1974|loc=Proposition 1.2}}  (Here, ''locally compact'' is in the sense of condition (3) in the corresponding article, namely each point has a local base of compact neighborhoods.)
* The product of a CG-2 space and a [[locally compact Hausdorff]] space is CG-2.{{sfn|Strickland|2009|loc=Proposition 2.6}}{{sfn|Rezk|2018|loc=Proposition 7.5}}

When working in a [[category (mathematics)|category]] of compactly generated spaces (like all CG-1 spaces or all CG-2 spaces), the usual [[product topology]] on <math>X\times Y</math> is not compactly generated in general, so cannot serve as a [[categorical product]].  But its k-ification <math>k(X\times Y)</math> does belong to the expected category and is the categorical product.{{sfn|Lamartin|1977|loc=Proposition 1.11}}{{sfn|Rezk|2018|loc=section 3.5}}

===Continuity of functions===

The continuous functions on compactly generated spaces are those that behave well on compact subsets.  More precisely, let <math>f:X\to Y</math> be a function from a topological space to another and suppose the domain <math>X</math> is compactly generated according to one of the definitions in this article.  Since compactly generated spaces are defined in terms of a [[final topology]], one can express the [[continuity (topology)|continuity]] of <math>f</math> in terms of the continuity of the composition of <math>f</math> with the various maps in the family used to define the final topology.  The specifics are as follows.

If <math>X</math> is CG-1, the function <math>f</math> is continuous if and only if the [[restriction (mathematics)|restriction]] <math>f\vert_K:K\to Y</math> is continuous for each compact <math>K\subseteq X.</math>{{sfn|Willard|2004|loc=Theorem 43.10}}

If <math>X</math> is CG-2, the function <math>f</math> is continuous if and only if the [[composition (functions)|composition]] <math>f\circ u:K\to Y</math> is continuous for each compact Hausdorff space <math>K</math> and continuous map <math>u:K\to X.</math>{{sfn|Strickland|2009|loc=Proposition 1.11}}

If <math>X</math> is CG-3, the function <math>f</math> is continuous if and only if the restriction <math>f\vert_K:K\to Y</math> is continuous for each compact Hausdorff <math>K\subseteq X.</math>

===Miscellaneous===

For topological spaces <math>X</math> and <math>Y,</math> let <math>C(X,Y)</math> denote the space of all continuous maps from <math>X</math> to <math>Y</math> topologized by the [[compact-open topology]].  If <math>X</math> is CG-1, the [[path component]]s in <math>C(X,Y)</math> are precisely the [[homotopy]] equivalence classes.{{sfn|Willard|2004|loc=Problem 43J(1)}}