Why finite dimensional MCS-space implies $\mathbb{R}^m\times cone$ locally?

Question

Frequently, I see a statment of the result in reference that a finite dimensional Alexandrov spaces is locally a product $\mathbb{R}^m\times cone$. It seems that this result comes from Perelman's paper "Elements on Morse theory on Alexandrov spaces". In fact Perelman proved that a finite dimensional Alexandrov spaces is an MCS-spaces(MCS=multiple conic singularities). How it imples that result? Maybe it is easy, but I don't know how.

Def. A metrizable space $M_n$ is a MCS-space of dimension $n$ if, for each $x\in M_n$, there exists a neighborhood $U$ of $x$ and an compact MCS-space $M_{n-1}$ of dimension $n − 1$, such that there exists a homeomorphism from the cone of $M_{n-1}$ to $U$ such that it sends the cone point to $x$. Here, we regard the case of -1 dimension as the empty set and its cone as a single point.

By the way, in the remark of the definition of MCS-space in the paper above. Let subset $A^n$ consists points $x\in M_n$ whose conic nbhd admits a topological splitting $\mathbb{R}^m\times C(M_{n-m-1})$ (here $C(M_{n-m-1})$ is a cone over an $n-1$ dimensional MCS-space) for all $m\leq l$. Perelman said that $A^l$ is an $l$-manifold. I don't know why. In my opinion, $A^l-A^{l-1}$ is an $l$-manifold but not $A^l$.

The questions overlap https://math.stackexchange.com/questions/1130005/. Since no one answered there, I post it here.

Thanks!

Answer 1

I'm not sure whether it's generally the case that when you split off an $\mathbb{R}^m$ factor from a cone on an MCS space, the other factor is also a cone on an MCS space. Except, of course, if $m=0$.

The way it works for Alexandrov spaces is that the proof that the space is MCS involves finding some regular functions near the point. If you find $m$ of them, you have a fibration so that the space is locally $\mathbb{R}^m \times F$, for some fiber $F$. Perelman showed you could always find another function, which has a unique maximum on each fiber, and is regular elsewhere. This function shows $F$ is a cone. That it is a cone on an MCS space follows inductively.

For the other question, the definition of $A^l$ should include a statement that there is no such splitting for $m > l$. That way $A^{l-1}$ is not contained in $A^l$.