Topological Manifolds

Let us recall that a topological space M is a topological manifold if M is second-countable Hausdorff and locally Euclidean, i.e. each point has a neighborhood that is homeomorphic to an open ball of E n for some n. However, if we would like to consider a topological manifold with a boundary, we have to extend this definition. Therefore, we introduce here the concept of a locally Euclidean space that covers both cases (with and without a boundary), i.e. where each point has a neighborhood that is homeomorphic to a closed ball of En for some n