Editing Compactly generated space (section)

===General framework for the definitions===

Let <math>(X,T)</math> be a [[topological space]], where <math>T</math> is the [[topological space#topology|topology]], that is, the collection of all open sets in <math>X.</math>

There are multiple (non-equivalent) definitions of ''compactly generated space'' or ''k-space'' in the literature.  These definitions share a common structure, starting with a suitably specified family <math>\mathcal F</math> of continuous maps from some compact spaces to <math>X.</math>  The various definitions differ in their choice of the family <math>\mathcal F,</math> as detailed below.

The [[final topology]] <math>T_{\mathcal F}</math> on <math>X</math> with respect to the family <math>\mathcal F</math> is called the '''k-ification''' of <math>T.</math>  Since all the functions in <math>\mathcal F</math> were continuous into <math>(X,T),</math> the k-ification of <math>T</math> is [[finer topology|finer]] than (or equal to) the original topology <math>T</math>.  The open sets in the k-ification are called the '''{{visible anchor|k-open|k-open set|text=k-open sets}}''' in <math>X;</math> they are the sets <math>U\subseteq X</math> such that <math>f^{-1}(U)</math> is open in <math>K</math> for every <math>f:K\to X</math> in <math>\mathcal F.</math>  Similarly, the '''{{visible anchor|k-closed|k-closed set|text=k-closed sets}}''' in <math>X</math> are the closed sets in its k-ification, with a corresponding characterization.  In the space <math>X,</math> every open set is k-open and every closed set is k-closed.  The space <math>X</math> together with the new topology <math>T_{\mathcal F}</math> is usually denoted <math>kX.</math>{{sfn|Strickland|2009|loc=Definition 1.1}}

The space <math>X</math> is called '''compactly generated''' or a '''k-space''' (with respect to the family <math>\mathcal F</math>) if its topology is determined by all maps in <math>\mathcal F</math>, in the sense that the topology on <math>X</math> is equal to its k-ification; equivalently, if every k-open set is open in <math>X,</math> or if every k-closed set is closed in <math>X;</math> or in short, if <math>kX=X.</math>

As for the different choices for the family <math>\mathcal F</math>, one can take all the inclusions maps from certain subspaces of <math>X,</math> for example all compact subspaces, or all compact Hausdorff subspaces.  This corresponds to choosing a set <math>\mathcal C</math> of subspaces of <math>X.</math>  The space <math>X</math> is then ''compactly generated'' exactly when its topology is [[coherent (topology)|coherent]] with that family of subspaces; namely, a set <math>A\subseteq X</math> is open (resp. closed) in <math>X</math> exactly when the intersection <math>A\cap K</math> is open (resp. closed) in <math>K</math> for every <math>K\in\mathcal C.</math>  Another choice is to take the family of all continuous maps from arbitrary spaces of a certain type into <math>X,</math> for example all such maps from arbitrary compact spaces, or from arbitrary compact Hausdorff spaces.

These different choices for the family of continuous maps into <math>X</math> lead to different definitions of ''compactly generated space''.  Additionally, some authors require <math>X</math> to satisfy a separation axiom (like [[Hausdorff space|Hausdorff]] or [[weak Hausdorff]]) as part of the definition, while others don't.  The definitions in this article will not comprise any such separation axiom.

As an additional general note, a sufficient condition that can be useful to show that a space <math>X</math> is compactly generated (with respect to <math>\mathcal F</math>) is to find a subfamily <math>\mathcal G\subseteq\mathcal F</math> such that <math>X</math> is compactly generated with respect to <math>\mathcal G.</math>  For coherent spaces, that corresponds to showing that the space is coherent with a subfamily of the family of subspaces.  For example, this provides one way to show that locally compact spaces are compactly generated.

Below are some of the more commonly used definitions in more detail, in increasing order of specificity.

For Hausdorff spaces, all three definitions are equivalent.  So the terminology '''{{visible anchor|compactly generated Hausdorff space}}''' is unambiguous and refers to a compactly generated space (in any of the definitions) that is also [[Hausdorff space|Hausdorff]].