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(adsbygoogle = window.adsbygoogle || []).push({}); Rn (theRxR...xRcartesian 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 inEn (euclidian n-space) is used to assume the hausdorff and countable basis (paracompact) requirements. The hausdorff assumption, I have no trouble with, asRn clearly meets that condition as I understand it above. But I don't assume thatEn orRn have a countable basis of open sets.

Can someone prove to me thatRn has a countable basis of open sets?

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# Question on the properties of a manifold

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