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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. It is actually even the space of execution paths of the directed segment.

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.

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. It is actually the space of execution paths of the directed segment.

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. It is actually even the space of execution paths of the directed segment.

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.

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. It is actually the space of execution paths of the directed segment.