Delta-generated spaces vs CW complexes

Question

$\newcommand\Top{\mathrm{Top}}\newcommand\CW{\mathrm{CW}}\newcommand\Deltagenerated{\text{$\Delta$-generated}}\newcommand\Spaces{\mathrm{Spaces}}\newcommand\DeltaSpaces{\text{$\Delta$-Spaces}}$I am trying to understand the difference between Delta-generated spaces and CW complexes. Delta-generated spaces are colimits of the $n$-simplices $\Delta^n$ in $\Top$; here we take ordinary colimits in $\Top$, not homotopy colimits. CW complexes are special types of colimits of $\Delta^n$; namely, they are $X = \varinjlim X_i$, where $X_{k+1}$ is the pushout of $X_k$ along the inclusion map $\coprod \ \partial \Delta^n \rightarrow \coprod \Delta^n$. One important point seems to be that all $n$-cells are attached simultaneously.

By the above discussion, there is an inclusion $\CW \hookrightarrow \Deltagenerated$. This inclusion is not an equivalence since there are $\Delta$-generated spaces that are not homeomorphic to CW-complexes. I think one example is the colimit $[0,1]^\text{dis}\times [0,1] \leftarrow [0,1]^\text{dis} \rightarrow [0,1]$. Here $[0,1]^\text{dis}$ is the interval equipped with the discrete topology, $[0,1]^\text{dis}\rightarrow [0,1]$ is the identity map, and $[0,1]^\text{dis}\times [0,1] \leftarrow [0,1]^\text{dis}$ is induced by $[0,1] \leftarrow 0$. The issue is that the “1-cells” are not attached simultaneously.

However, it is not clear to me whether any $\Delta$-generated space is homotopy equivalent to a CW complex. The above example is contractible and hence homotopy equivalent to the point.

Question: is any $\Delta$-generated space homotopy equivalent to a CW complex? Alternatively, does Whitehead's theorem hold for $\Delta$-generated spaces?

I suspect that the answer to both of these questions is no (just because there must be a good reason that we use CW complexes) but I would like to know an explicit example.

A related point is that the inclusion $\CW \hookrightarrow \Spaces$ is not closed under (ordinary) colimits, and the smallest subcategory of $\Spaces$ containing $\CW$ is $\DeltaSpaces$. However, the inclusion of $\infty$-categories $\CW \hookrightarrow \Spaces$ does preserve homotopy colimits; see Example 6.2.2.9 https://kerodon.net/tag/02F5. However, the inclusion of the ordinary category of spaces into the $\infty$-category of spaces does not send ordinary colimits to homotopy colimits.

Edit See Proposition 3.11 of The D-Topology for Diffeological Spaces, https://arxiv.org/abs/1302.2935, for the proof that every locally path-connected first countable topological space is $\Delta$-generated, as stated in the answer below.

Answer 1

The category of $\Delta$-generated spaces is quite large as it includes all first countable, locally path connected spaces. Hence, all compact, connected, locally connected subsets of $\mathbb{R}^n$ are $\Delta$-generated. This is includes the infinite earring space, Sierpiński carpet, Menger cube, etc. In particular, the infinite earring space $\mathbb{E}$ is not homotopy equivalent to any CW-complex because the existence of a "wild point" (point at which a space is not semilocally simply connected) is a homotopy invariant.

Let $X$ be a CW-approximation to $\mathbb{E}$ and $f:X\to \mathbb{E}$ be a weak homotopy equivalence. Then as noted above, this cannot be a homotopy equivalence. Thus the Whitehead Theorem does not hold in the category of $\Delta$-generated spaces.