# Compact-open topology and Delta-generated spaces

Consider the set of continuous maps $$C^0([0,1],[0,1])$$ equipped with the compact-open topology. It is metrisable, and therefore sequential. It is also a k-space: see http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf Proposition 1.6. The proof relies on the facts that every k-closed subset is in particular $$\overline{\mathbb{N}}$$-closed where $$\overline{\mathbb{N}}$$ is the one-point compactification of $$\mathbb{N}$$, that every $$\overline{\mathbb{N}}$$-closed subset is sequentially closed, and therefore the kelleyfication functor adds no open subsets in the topology. Since $$\overline{\mathbb{N}}$$ is not $$\Delta$$-generated (its $$\Delta$$-kelleyfication is the discrete space $$\overline{\mathbb{N}}^\delta$$), the preceding proof does not work for $$\Delta$$-generated spaces.

I am (almost) sure that $$C^0([0,1],[0,1])$$ is not $$\Delta$$-generated and I would appreciate to see a proof.

Motivation: This question is important for me because I am trying to understand specific things about the topology of the space of execution paths of a cellular multipointed $$d$$-space having a finite number of cells (in the sense of https://arxiv.org/abs/1904.04159). And the space above appears everywhere.

• I have a (maybe incorrect) note that say that $C(I,I)$ (compact-open topology) is uniformly locally contractible. In particular it is locally path-connected, so should be $\Delta$-generated since it is first-countable. Feel free to correct me on this. – Tyrone Jan 8 at 13:00
• @Tyrone I don't understand your argument. – Philippe Gaucher Jan 8 at 13:12
• Every locally path-connected first-countable space is $\Delta$-generated. I learned this from one of Dan Christensen's papers. – Tyrone Jan 8 at 13:19
• @Tyrone Do you have a reference please ? I will accept that as an answer by the way. – Philippe Gaucher Jan 8 at 13:20
• It's Proposition 3.11 in The D-topology for Diffeological spaces. I don't have a proof that $C(I,I)$ is locally path-connected to hand, but that doesn't seem to hard to sort out. If you're happy I'll post an answer. – Tyrone Jan 8 at 13:23

The mapping space $$C([0,1],[0,1])$$ in the compact-open topology is in fact $$\Delta$$-generated.
The reason for this is that every locally path-connected first-countable space is $$\Delta$$-generated. This was proved by Christensen, Sinnamon, and Wu in Proposition 3.11 of their paper The D-Topology for Diffeological Spaces, Pacific J. Math., 272, (2014). As has already been noted, $$C([0,1],[0,1])$$ is metrisable, and hence first-countable. In addition it's not difficult to see that it is also locally path-connected (in fact it is locally contractible in a strong sense).