Let $X$ be a topological space. The simplicial set $\operatorname{sing}(X)$ has as its $n$ simplicies all singular $n$-simplices $\Delta^n \to X$. The realization of the singular simplicial set is then $$ \lvert\operatorname{sing}(X)\rvert = \left.\coprod_{n\in\mathbb{N}}\underbrace{\operatorname{sing}_n(X)}_{\text{discrete}} \times \Delta^n\middle/ \left(\underset{\substack{\sigma\colon \Delta^n\to X \\ p \in \Delta^k \\ g\colon [0..k] \to [0..n] \text{ nondecreasing}}}{(g^*(\sigma), p) \sim (\sigma, g_*(p))}\right)\right., $$ where $g_*\colon \Delta^k \to \Delta^n$ is the affine map sending the $i$th vertex to the $g(i)$th vertex, and $g^*(\sigma) = \sigma \circ g_*$.
Let $x \in X$. For each $\sigma\colon\Delta^n \to X$ and $p \in \Delta^n \setminus \partial\Delta^n$ such that $\sigma(p) = x$, the point $[(\sigma, p)]$ is only identified with degerate higher-degree copies of $\sigma$, but not any boundary maps of nondegenerate simplices because $p$ is in the interior. We thus get many points $[(\sigma, p)]$ in $\lvert \operatorname{sing}(X)\rvert$ corresponding to $x \in X$, and so we should not expect there to be a homeomorphism $X \cong \lvert \operatorname{sing}(X)\rvert$.
We could define a map $\varepsilon\colon\lvert\operatorname{sing}(X)\rvert \to X$ by $[(\sigma, p)] \mapsto \sigma(p)$, well-defined because $g^*(\sigma)(p)=\sigma(g_*(p))$, and this map is the counit of the adjuction $|\cdot| \dashv \operatorname{sing}$ and is a homotopy equivalence, but it is not typically a homeomorphism except maybe when $X$ is discrete.
Now I'll write $\operatorname{topsing}(X)$ for the simplicial space whose space of maps is the space of maps $\Delta^n \to X$ with the compact-open topology. I believe the realization should then be analogous: $$ \lvert\operatorname{topsing}(X)\rvert = \left.\coprod_{n\in\mathbb{N}}\operatorname{topsing}_n(X) \times \Delta^n\middle/ \left(\underset{\substack{\sigma\colon \Delta^n\to X \\ p \in \Delta^k \\ g\colon [0..k] \to [0..n] \text{ nondecreasing}}}{(g^*(\sigma), p) \sim (\sigma, g_*(p))}\right)\right., $$
Similar to before, I believe the open set $\operatorname{topsing}_n(X) \times (\Delta^n \setminus \partial \Delta^n)$ in the coproduct avoids nondegenrate identification in the quotient, so we still should not expect a homeomorphism $X \cong \lvert\operatorname{topsing} X\rvert$. I believe we could still define $\varepsilon'\colon |\operatorname{topsing}(X)| \to X$ similarly.
My main questions:
- If $\varepsilon\colon\lvert\operatorname{sing}(X)\rvert \to X$ is a homeomorphism, does it follow that $X$ is discrete?
- If $\varepsilon'\colon |\operatorname{topsing}(X)| \to X$ is a homeomorphism, does it follow that $X$ is discrete?
- Is there another kind of "singular everything-complex" whose realization does capture the homeomorphism type of nice spaces? (say, manifolds; I don't care about examples like $\mathbb{Q}$.) I suppose one answer is the "all-continuous-maps-between-simplices-complex", presheafs on the full subcategory on $\{\Delta^0, \Delta^1, \dots\} \subset \mathsf{Top}$ in which the quotient identifies points (singular zero-simplices) with all of the corresponding points in other singular simplices. But is there something natural that round-trips nice homeomorphism types with less data than this?