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I engaged two definitions for a compactly generated space:

http://en.wikipedia.org/wiki/Compactly_generated_space

1) In topology, a compactly generated space (or k-space) is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space $X$ is compactly generated if it satisfies the following condition: A subspace $A$ is closed in $X$ if and only if $A\cap K$ is closed in $K$ for all compact subspaces $K\subseteq X$.

http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf

2) A subset $Y\subseteq X$ is $k$-closed if $u^{-1}\left(Y\right)$ is closed in $K$ for every compact Hausdorff space $K$ and every continuous map $u:K\rightarrow X$. These sets can be recognized as the closed sets of a topology (finer than the original topology) and we say that $X$ is compactly generated if this topology is not properly finer than the original topology.

Question: are these definitions equivalent? And if not then wich is the most usual and or convenient to practicize?

drhab
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  • They are not equivalent: Let's call the later spaces k-spaces. Since compactly generated spaces have the final topology with respect to the maps from compact spaces to $X$, and $k$-spaces have the final topology for all maps from compact Hausdorff spaces to $X$, it follows that every $k$-space is compactly generated. For a compactly generated space which is not a $k$-space, see $\Bbb Q^\times\Bbb Q^$, the square of the one-point compactification of $\Bbb Q$. – Stefan Hamcke Feb 21 '15 at 17:00
  • In my experience the Wikipedia definition is the usual one, though some people add the requirement that the space be Hausdorff; I don’t think that I’ve seen Strickland’s before. However, I don’t do algebraic topology, and I don’t care for category theory, so if it’s useful primarily in that context, it’s not surprising that I’ve not encountered it. As Stefan says, they’re not equivalent in general, though they are equivalent in Hausdorff spaces. – Brian M. Scott Feb 21 '15 at 17:35
  • @StefanHamcke Thank you. My suspicion is now confirmed by your comment. I dislike this sort of situations. – drhab Feb 21 '15 at 17:36
  • @BrianM.Scott Thank you as well. I will have to learn to live with this :( – drhab Feb 21 '15 at 17:37

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(With apologies for introducing another term, I will attempt to avoid ambiguity by calling the spaces defined in wikipedia "quasicompactly-generated", and the spaces defined in Strickland's notes "$k$-spaces". Fortunately we're not also considering Kelley's definition, so I don't need a third term! More discussion of the different possible definitions can be found here.)

As discussed in the comments, these two definitions are not equivalent. As for usefulness, I'm not aware of how these spaces are used outside of algebraic topology (but it would be nice if someone could explain another use of these spaces!). The $k$-spaces are the ones which are used in algebraic topology. The principal reason they are used is that $k$-spaces form a cartesian closed category, which means that spaces of continuous maps behave nicely. This implies such pleasant facts as: if $q: X \to Y$ is a quotient map between $k$-spaces, then $\mathrm{Id}_Z \times q : Z \times X \to Z \times Y$ is a quotient map, when the products are taken in the category of $k$-spaces. I see no reason to expect the quasicompactly-generated spaces to be cartesian closed, although I don't know for sure that they're not.

Note moreover that algebraic topologists usually impose a weak Hausdorff condition on their $k$-spaces, as discussed in the Strickland notes. For some discussion of why this weak Hausdorff condition is used rather than Hausdorffness, see here and here.

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