"Locally compact"-ly generated topological spaces

Question

Let $P$ be a property of topological spaces - here I am interested in "compact" and "locally compact".

A topological space $X$ is $P$-ly-generated if, for any topological space $Y$, a function $f : X \rightarrow Y$ is continuous if and only if $f \circ i$ is continuous for any topological space $K$ such that $P$ and any continuous function $i : K \rightarrow X$.

An important result due to Martin Hyland says that the exponentiable objects in locales are the locally compact locales.

My question is, is compact-ly-generated equivalent to locally-compact-ly-generated?

My second question is whether anyone knows the 1979 reference to the mentioned result of Martin Hyland.

Answer 1

If $X$ is locally compactly generated then $X$ is compactly generated because every locally compact space is compactly generated.

So given $f:X \to Y$ a map such that $f\circ i$ is continuous for every compact $i:K \to X$, then it follows that for every locally compact $j:Q \to X$ one can test the continuity of $f \circ j$ on all compact $K \to Q \to X$, hence $f \circ j$ is continuous, and as $X$ is locally compactly generated it follows that $f$ is continuous.

The problem is actually with the converse: If "compact" means "compact hausdorff" then every compact space is locally compact so the converse hold trivially, but this fails if compact means quasi-compact.

Indeed let $X$ be a non-compactly generated space (in particular it is not locally compactly generated).

Let $X^*$ be the space obtained by adding one point $*$ to $X$, with the open subset of $X^*$ being the open subset of $X$, together with $X^*$ itself.

$X^*$ is quasi-compact because a collection of open is covering if and only if it contains $X^*$ itself. But $X^*$ is not locally compactly generated: Given a map $F:X \to Y$ which is continuous when restricted to any (locally) compact space but continuous. Then we can show that the induced map $X^* \to Y^*$ will be continuous when restricted to any locally compact space, but not continuous itself, so $X^*$ is compact but not locally compactly generated.

In the last claim, I'm using that given a continuous map $j: Q \to X^*$ the preimage $Q' = j^*X$ is a locally compact space with a map to $X$ and the map $Q \to X^*$ can be factored as $Q \to Q'^* \to X^*$ with $(Q')^*$ being locally compact.