(4.1.2) A topological space $T$ is called an $n$-dimensional pseudo-manifold with boundary if it admits a triangulation $\mathscr{T}$ satisfying the following conditions:

1. (1)

   (dimensional homogeneity) $T=|\mathscr{T}|$ is the union of all $n$-simplices.

2. (2)

   (non-branching) Every $(n-1)$-simplex is a face of precisely one or two $n$-simplices.

3. (3)

   (strong connectedness) For every pair of $n$-simplices $\sigma$ and $\sigma^{\prime}$ in $\mathscr{T}$, there is a sequence of $n$-simplices

   $$\sigma=\sigma_{0},\sigma_{1},\ldots,\sigma_{\ell}=\sigma^{\prime}$$

   such that the intersection $\sigma_{i}\cap\sigma_{i+1}$ is an $(n-1)$-simplex for all $i$.

We say that $T$ is a closed pseudo-manifold if we can replace condition (2) by the property that every $(n-1)$-simplex is a face of precisely two $n$-simplices. A typical example of a 2-dimensional closed pseudo-manifold which is not a manifold is the pinched torus.