This is a question related to another one in MO
Background : A special simplicial space $X_{\cdot}$ is a simplicial space with $X_{0}=\ast$ and $X_{n}\simeq X_{1}^{n}$ via the simplicial map $v_{i}:[1]\rightarrow [n]$ where
$v_{i}(1)=i$ and $v_{i}(0)=i-1$.
Denote the category of special simplicial spaces by $SsTop$ and the category of pointed spaces $Top_{\ast}$ (i,e.the pointed compactly generated weak Hausdorff spaces).
Now $\Omega_{\cdot}$, being defined via $$\Omega_{k}X\equiv hom((\triangle^{k},(\triangle^{k})^{0}),(X,x_{0})),$$ is a right adjoint of the realization functor $\vert-\vert$, where $(\triangle^{k})^{0}$ means the vertices of $\triangle^{k}$ (face and degeneracy defined as in the singular simplicial space). We call $\Omega_{\ast}X$ the special singular simplicial space of $X$, for $X\in Top_{\ast}$.
We thus obtain $$\vert-\vert: SsTop\leftrightarrows Top_{\ast}: \Omega$$
Now my first question is: Do we have $\vert\Omega_{\cdot}X\vert\rightarrow X$ is a (weak) homotopy equivalence? Or under what condition of $X$, one can have this?
This has been actually proved in [May: Classifying spaces and fibrations, Porp. 15.5], under the assumption that $X$ is well-pointed. However there some properness assumption of $\Omega_{\cdot}X$ is made use of, but which is not quite clear to me.
More precisely, he first indicated there is a simplicial map from $B_{\cdot}\Lambda X$, the bar construction of the Moore loop space $\Lambda X$ which is a topological monoid, to $\Omega_{\cdot}X$ and it is a level-wise homotopy equivalence as both simplicial spaces are special and the simplicial map induces a homotopy equivalence on the $1$-simplices.
Then by applying the lemma:
If a simplicial map between two proper simplicial spaces is level-wise homotopy equivalence, then it induces a homotopy equivalence after applying realization functor.
and using the homotopy equivalence $\vert B_{\cdot}\Lambda X\vert \rightarrow X$ as well as the decomposition $$\vert B_{\cdot}\Lambda X\vert\rightarrow \vert\Omega_{\cdot}X\vert\rightarrow X,$$ the assertion follows.
But in applying this lemma, we need to have $\Omega_{\cdot}X$ and $B_{\cdot}\Lambda X$ are proper simplicial spaces. For the latter it is rather fine (see below), but for the former it is then not so clear.
So my second question is: How to see $\Omega_{\cdot}X$ is proper when X is well-pointed? Or it is actually not the case?
For the bar construction of the Moore loop space, $B_{\cdot}\Lambda X$, I think I can see it is proper as the Moore loops space is well-pointed if $X$ is. For $\Omega_{\cdot}X$, though, it is not so clear to me. Beside, as far as I can see, one also cannot apply the approach in the solution of the MO question cited above as well as the construction of the counterexample there.
Any comment or solution will be very much appreciated.
Thank you in advance.
