I am reading Burago, Burago and Ivanov's book A course in metric geometry. In chapter 10 the mention that Alexandrov spaces of curvature bounded below have a stratification into topological manifolds. On further reading, I found that on Perelman's article Elements of Morse theory on Alexandrov spaces that the stratification comes from his result that says that an Alexandrov space is a Multiple Conical Singularities Space (MCS). The stratification is as follows:
Let $X$ be an Alexandrov space. The $l$-dimensional stratum is the subset of points $x\in X$ whose conical neighborhood admit a topological splitting $\mathbb{R}^m \times \mathrm{Cone}(\Sigma_{n-m-1})$, where $\Sigma_{n-m-1}$ is a compact MCS-space of dimension $n-m-1$ for all $m\leq l$.
In Burago's book the mention that it is possible to prove that there is no codimension $2$ stratum for Alexandrov spaces.
This implies that the set of topologically singular points (i.e. points whose space of directions is not homeomorphic to the sphere) which are not boundary points, has codimension at least $3$.
Could you help me prove this?
I have tried to give a proof by induction on the dimension of $X$ but I am finding some difficulties. The base of the induction would be clear since $1$ and $2$ dimensional Alexandrov spaces are topological manifolds. Then I assume that the affirmation is true for every Alexandrov space of dimension $\leq n$. Now if $X$ has dimension $n+1$, the singular points are not in the top-dimensional stratum since the said stratum is the set of topologically regular points. Therefore it has at least codimension 1. Further if we exclude boundary points, since the closure of the $n-1$ stratum is the boundary, we obtain that the set of topologically singular points which are not on the boundary is at least $2$. Here is were I find trouble. I can't seem to be able to use my induction hypothesis. My intuition tells me that essentially I have to use that the space of directions is an $n$-dimensional Alexandrov space. Could you give me a hint?
