- 12

- 0

**R**n (the

**R**x

**R**...x

**R**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

**E**n (euclidian n-space) is used to assume the hausdorff and countable basis (paracompact) requirements. The hausdorff assumption, I have no trouble with, as

**R**n clearly meets that condition as I understand it above. But I don't assume that

**E**n or

**R**n have a countable basis of open sets.

Can someone prove to me that

**R**n has a countable basis of open sets?