Editing Compactly generated space (section)

===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.