1. Every manifold has a natural topology since it is locally Euclidean . 2. The property of being locally Euclidean is preserved by local homeomorphisms. 3. This much is a fragment of a typical locally Euclidean topological group. 4. In particular, being locally Euclidean is a topological property. 5. It is true, however, that every locally Euclidean space is T 1. 6. A "'topological manifold "'is a locally Euclidean Hausdorff space. 7. An example of a non-Hausdorff locally Euclidean space is the line with two origins. 8. The definition I follow is the " locally Euclidean " one so I allow the long line to be a manifold. 9. In the previous section, a surface is defined as a topological space with certain properties, namely Hausdorff and locally Euclidean . 10. There are also topological manifolds, i . e ., locally Euclidean spaces, which possess no differentiable structures at all.
