Let \(X\) be a CW complex whose \(n\)-skeleton \(X^n\) is obtained from \(X^{n-1}\) by attaching \(n\)-disks \(D^n_\alpha\) along attaching maps \(\varphi_\alpha\colon S^{n-1}\to X^{n-1}\). The characteristic map of \(D^n_\alpha\) is the composite \(\Phi_\alpha\colon D^n_\alpha\hookrightarrow X^{n-1}\sqcup\coprod_\beta D^n_\beta\to X^n\hookrightarrow X\), whose restriction to \(S^{n-1}\) is \(\varphi_\alpha\). The \(n\)-cell \(e^n_\alpha\) is the image \(\Phi_\alpha(D^n\sm S^{n-1})\).
Each \(n\)-cell \(e^n_\alpha\) is homeomorphic to the open disk \(D^n\sm S^{n-1}\cong\RR^n\), and the underlying set of \(X\) is the disjoint union of the cells \(e^n_\alpha\) over all \(n\) and \(\alpha\).
See [@Hat02].