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](http://austinmohr.com/home/?page_id=146) nor the literature I have consulted (in particular [Counterxamples in Topology](http://books.google.de/books/about/Counterexamples_in_Topology.html?id=DkEuGkOtSrUC&redir_esc=y)) 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](http://austinmohr.com/home/?page_id=146) 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). 



Tietze's extension theorem for compact subspaces
The answer to this question give an example of a topological space which is compactly generated, but is not functionally Hausdorff.


[Is a compactly generated Hausdorff space functionally Hausdorff?](https://mathoverflow.net/questions/39670/is-a-compactly-generated-hausdorff-space-functionally-hausdorff)


