The topological question:

Are there Hausdorff topological spaces $X$ which are compactly generated (=Kelly spaces = $k$-spaces, that is, a subset is closed if its intersection with every compact set is closed) which fail to satisfy Tietze's extension theorem for compact subsets (that is, there is a real-valued continuous function defined on some compact subset without continuous extension to the whole space)?

A possibility to answer this question would be to exhibit a compactly generated space which is not *completely Hausdorff* (continuous functions do not separate points).
Unfortunately, neither the space book nor the literature I have consulted (in particular Counterxamples in Topology) contained an answer.

I have chosen the Functional Analysis tag because such a space would yield a very natural example of a projective (or inverse) limit of Banach spaces $C(K)$ ($K$ compact) with surjective linking maps $C(L)\to C(K)$, $f\to f|_K$ for $K\subseteq L$ (recall that compact spaces are normal so that Tietze's theorem applies) such that the canonical map from the projective limit (which is $C(X)$ because of compact generation) to the "steps" is not surjective.

Edit. Recalling the implications *completely Hausdorff* $\Rightarrow$ $T_{2\, 1/2}$
and *second countable* $\Rightarrow$ *compactly generated* one gets a list of 6 examples from spacebook namely

Double Origin Topology

Irrational Slope Topology

Minimal Hausdorff Topology

Prime Integer Topology

Relatively Prime Integer Topology

Simplified Arens Square (this one is particularly simple).