I'm reading the article Orientability and fundamental classes of Alexandrov spaces with applications by Ayato Mitsuishi.
There are the following definitions: for $n\geq 1$, an $n$-dimensional MCS space (standing for 'multiple conic singularities') is a separable metrizable space $X$ with the property that for any $x\in X$ there is a $(n-1)$-dimensional compact MCS space $\Sigma$ and a nbhd $U$ such that $(U,x)$ and $(c(\Sigma),o)$ are pointed homeomorphic, where $c(\Sigma) : = \Sigma\times [0,\infty)/\Sigma\times \{0\}$ is the open cone generated by $\Sigma$ and $o$ is the apex of the cone. A $0$-dimensional MCS space is taken to be a discrete space. For example, every $n$-dimensional manifold is a $n$-dimensional MCS space.
It is known that the subset $X_\mathrm{top}$ of $n$-manifold points in $X$ is an open dense subspace. Its complement, denoted by $S$, is the set of singular points.
I've been struggling with the existence of an exact sequence $$ H^{n-1}(S)\to H^n_\mathrm{c}(X_\mathrm{top})\to H^n_\mathrm{c}(X)\to H^n(S) $$ where $H^n$ denotes the Alexander-Spanier cohomology and subscript $\mathrm{c}$ denotes compactly supported cohomology. I simply don't understand where this sequence may come from or wheter it is some kind of property of Alexander-Spanier cohomology in general; honestyl I'm not familiarized with this cohomology theory so any insight or good reference will be appreciated.