Is the Moore Loop space of a well-pointed space well-pointed?

Question

Suppose that $X$ is a well-pointed topological space (i.e. the basepoint is cofibered, not necessarily closed). Is the Moore loop space/space of measured looks $\Omega_MX$ well-pointed? I have seen this claimed in multiple places, but have not been able to locate or produce a proof.

To recap, the Moore loop space is the subspace $\Omega_MX$ of $X^{[0,\infty)}\times[0,\infty)$ consisting of pairs $(\gamma,t)$ s.t. $\gamma(0)=\ast$ and $\gamma(s)=\ast$ for $s\ge t$, topologized as subspace of the product where the first factor carries the compact-open topology. This space is pointed with respect to $(c_{\ast},0)$, where $c_{\ast}$ denotes the constant loop at the basepoint $\ast$ of $X$. This is, in fact, the identity element in the natural topological monoid structure on $\Omega_MX$ given by concatenating loops.

Intuitively, I want to proceed similarly to the usual loop space (where the result holds). To do so, we fix a Strøm structure $(\varphi,H)$ on $X$ and want to construct a Strøm structure $(\tilde{\varphi},\tilde{H})$ on $\Omega_MX$. The distance function could be $\tilde{\varphi}\colon\Omega_MX\rightarrow I,\,(\gamma,t)\mapsto t\lor\max_{u\in I}\varphi\gamma(u)$ (where $\lor$ denotes maximum). It is clear that $\tilde{\varphi}^{-1}(0)=\{(c_{\ast},0)\}$. The homotopy could look like $\tilde{H}\colon\Omega_MX\times I\rightarrow\Omega_MX,\,((\gamma,t),s)\mapsto(u\mapsto H(\gamma(u),s),?)$. However, it is not clear to me what the second component should look like. For $s=0$, it should be $?=t$, but if $s>\tilde{\varphi}(\gamma,t)$, then it should be $?=0$, and for the homotopy to be well-defined, $u\ge?$ has to imply $H(\gamma(u),s)=\ast$, most likely by virtue of implying $s>\varphi\gamma(u)$. Essentially, this comes down to estimating how much of the tail end of $\gamma$ lies in $\varphi^{-1}([0,s))$, and I'm not sure how to do this by a continuous map.

Answer 1

I believe that you are familiar with the book Homotopietheorie by tom Dieck, Kamps and Puppe.

Recall from that book that an inclusion $i:A\subseteq X$ is a weak cofibration (or Dold cofibration, or h-cofibration) if for each map $f:X\rightarrow Y$ and each homotopy $F:A\times I\rightarrow Y$ with $F_t=fi$ for $0\leq t\leq 1/2$ there is a homotopy $G:X\times I\rightarrow Y$ with $G_0=f$ and $G_ti=F_t$ for all $0\leq t\leq 1$.

Clearly every cofibration is a weak cofibration.

A pointed space $X$ is said to be weakly well-pointed (or h-well-pointed) if the basepoint inclusion $\ast\hookrightarrow X$ is a weak cofibration.

Theorem (Homotopietheorie, Satz 3.26, p.84) The following statements about a subspace inclusion $i:A\subseteq X$ are equivalent.

1. $i$ is a closed cofibration.
2. $i$ is a weak cofibration and $A$ is the zero set of a continuous function $\varphi:X\rightarrow [0,\infty)$. $\quad\blacksquare$

Now turn to the loop space $\Omega X$ and Moore loop space $\Omega_MX$ of a pointed space $X$.

Proposition (Homotopietheorie, Satz, p.180) If $X$ is a weakly well-pointed space, then $\Omega X$ and $\Omega_MX$ are weakly well-pointed. $\quad\blacksquare$

Being a special case of the above, $\Omega X$ and $\Omega_MX$ are weakly well-pointed when $X$ is well-pointed.

Now notice that the basepoint inclusion $\{(\ast,0)\}\subseteq \Omega_MX$ is the zero set of the map $\varphi:\Omega_MX\rightarrow[0,\infty)$ $$\varphi(\ell,t)= t.$$ It follows from the results above that $\Omega_MX$ is well-pointed whenever $X$ is weakly well-pointed.