Editing Compactly generated space (section)

===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.