Differentiable Manifolds and Mappings, Tangents, Vectorfields

Abstract

Note that Hausdorff is not a local property . Hence it cannot be dropped fomr definition . Take for example two copies of the real line and identity all pairs of points which have equal coordinates different from zero . The resulting space is locally Euclidean but not Hausdorff since the two origins do not have disjoint neighborhoods . To get an example for a locally Euclidean Hausdorff well-ordered set of half open intervals.