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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.

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  • $\begingroup$ 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. $\endgroup$ – Tyrone Jan 8 at 13:00
  • $\begingroup$ @Tyrone I don't understand your argument. $\endgroup$ – Philippe Gaucher Jan 8 at 13:12
  • $\begingroup$ Every locally path-connected first-countable space is $\Delta$-generated. I learned this from one of Dan Christensen's papers. $\endgroup$ – Tyrone Jan 8 at 13:19
  • $\begingroup$ @Tyrone Do you have a reference please ? I will accept that as an answer by the way. $\endgroup$ – Philippe Gaucher Jan 8 at 13:20
  • $\begingroup$ 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. $\endgroup$ – Tyrone Jan 8 at 13:23
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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).

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