Let \(n\geq0\). An \(n\)-manifold is a Hausdorff space \(M\) in which every point has an open neighborhood homeomorphic to \(\RR^n\) [@Hat02].
The definition of a topological manifold in [@Lee13] also requires \(M\) to be second-countable. The open long line is a Hausdorff space in which every point has an open neighborhood homeomorphic to \(\RR\), and it is not second-countable.