Any quotient of a compactly generated space is compactly generated

Question

I found a note about compactly generated. This is the article http://www.math.uiuc.edu/~franklan/Math535_1205.pdf.

I worry whether the proof of Proposition 2.4 is true. I not understand why the function $\tilde{q}$ from $X$ to $kY$ is continuous. I thought that the Proposition 2.4 is valid whenever $X$ is Hausdorff. But in that note, $X$ may be not a Haudorff space.

Can someone tell me?

Answer 1

That's what you could call the universal property of compactly generated spaces (I'll call them $c$-spaces for convenience):

If $X$ is a $c$-space and $Y$ is any space, then $f:X\to Y$ is continuous if and only if $f':X\to kY$, given by the same set map, is continuous.
Proof. Since the topology on $kY$ is finer than that on $Y$, one direction is clear. So assume $f:X\to Y$ is continuous. Since $X$ is a $c$-space, it suffices to show that $f'|_K:K\to kY$ is continuous whenever $K\subseteq X$ is compact. We know that $f|_K:X\to Y$ is continuous. But $Y$ and $kY$ have exactly the same continuous maps from compact spaces into them, so $f'|_K$ is continuous, and this means that $f'$ is.

Depending on how much category theory you know, you may understand this as the fact that the inclusion functor from the category of $c$-spaces to the category of all spaces has a right adjoint, the $k$-ification, with the counit being the identity $kY\to Y$.

Answer 2

If $Y$ is a Hausdorff space then $kY=Y$. Let $\tilde{q}=q$. Thus, $\tilde{q}$ is continuous