According to my text, a manifold should be 1) Hausdorff (that is t-2 separable, so there are disjoint open sets which are neighborhoods for any two points x and y), 2) locally euclidian (that there is a neighborhood U of a point x that is homeomorphic to an open subset U' of Rn (the RxR...xR cartesian product) and 3) has a countable basis of open sets. In most books, when the set out to illustrate something is a manifold, they usually explicitly show the locally euclidian character. Then the embedding of that manifold in En (euclidian n-space) is used to assume the hausdorff and countable basis (paracompact) requirements. The hausdorff assumption, I have no trouble with, as Rn clearly meets that condition as I understand it above. But I don't assume that En or Rn have a countable basis of open sets. Can someone prove to me that Rn has a countable basis of open sets?