How to define a compactly generated space? 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?

 A: (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.
