An (integral) affine manifold with singularities is a triple $(B,\Delta,\mathscr{A})$, where $B$ is a topological $n$-dimensional manifold, $\Delta\subset B$ a set which is locally a finite union of locally closed submanifolds of codimension at least $2$ and $\mathscr{A}$ is an (integral) affine structure on $B_{0}=B-\Delta$. A continuous map between (integral) affine manifolds with singularities

$$\alpha:B\rightarrow B^{\prime}$$

is (integral) affine if $\alpha^{-1}(B_{0}^{\prime})\cap B_{0}$ is dense in $B$ and the restriction $\alpha_{0}=\alpha|_{\alpha^{-1}(B_{0}^{\prime})\cap B_{0}}$:

$$\alpha_{0}:\alpha^{-1}(B_{0}^{\prime})\cap B_{0}\rightarrow B_{0}^{\prime}$$

is an (integral) affine map. We say that $\alpha$ is an (integral) affine isomorphism if $\alpha$ is an homeomorphism and $\alpha_{0}$ is an (integral) affine isomorphism of (integral) affine manifolds.