Question on the properties of a manifold

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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?
 

mathwonk

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try to prove it yourself. start with R. take the most natural countable set of center points i.e. the rationals., and take the most natural countable collection of open balls, i.e. rational radius intervals.

a piece of advice however:

these technical properties have no importance whatever, except that they are true; i.e. if you cannot prove them, but are willing to assume them, you are still going to get everything possible from the subject that is of importance.

of course i understand a curious poerson has difficulty by passing a puzzling statement.
 
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