Let \(M\) be an \(n\)-manifold and \(x\in M\). Then \(H_i(M, M\sm\ts{x};\ZZ)\cong\ZZ\) for \(i = n\) and \(H_i(M, M\sm\ts{x};\ZZ) = 0\) for \(i\neq n\). In particular, a nonempty space that is both an \(m\)-manifold and an \(n\)-manifold has \(m = n\).
Let \(M\) be a nonempty space that is an \(n\)-manifold for some \(n\geq 0\). The dimension of \(M\) is this integer \(n\), equivalently the unique \(i\) with \(H_i(M, M\sm\ts{x};\ZZ)\neq 0\) for any \(x\in M\).
See [@Hat02].