Editing Compactly generated space (section)

==K-ification==

Given any topological space <math>X</math> we can define a possibly [[finer topology]] on <math>X</math> that is compactly generated, sometimes called the '''{{visible anchor|k-ification}}''' of the topology. Let <math>\{K_\alpha\}</math> denote the family of compact subsets of <math>X.</math> We define the new topology on <math>X</math> by declaring a subset <math>A</math> to be closed [[if and only if]] <math>A \cap K_\alpha</math> is closed in <math>K_\alpha</math> for each index <math>\alpha.</math> Denote this new space by <math>kX.</math> One can show that the compact subsets of <math>kX</math> and <math>X</math> coincide, and the induced topologies on compact subsets are the same. It follows that <math>kX</math> is compactly generated. If <math>X</math> was compactly generated to start with then <math>kX = X.</math> Otherwise the topology on <math>kX</math> is strictly finer than <math>X</math> (i.e., there are more open sets).

This construction is [[functorial]].  We denote <math>\mathbf{CGTop}</math> the full subcategory of [[category of topological spaces|<math>\mathbf{Top}</math>]] with objects the compactly generated spaces, and <math>\mathbf{CGHaus}</math> the full subcategory of <math>\mathbf{CGTop}</math> with objects the Hausdorff spaces. The functor from <math>\mathbf{Top}</math> to <math>\mathbf{CGTop}</math> that takes <math>X</math> to <math>kX</math> is [[adjoint functors|right adjoint]] to the [[Subcategory#Formal definition|inclusion functor]] <math>\mathbf{CGTop} \to \mathbf{Top}.</math>

The [[exponential object]] in <math>\mathbf{CGHaus}</math> is given by <math>k(Y^X)</math> where <math>Y^X</math> is the space of [[continuous map]]s from <math>X</math> to <math>Y</math> with the [[compact-open topology]].

These ideas can be generalized to the non-Hausdorff case.{{sfn|Brown|2006|loc=section 5.9}} This is useful since identification spaces of Hausdorff spaces need not be Hausdorff.