no countable dense subset of a complete metric space is a
1) is a first category set:
(trivially). Suppose that . Then there is a ball aisolating the point. Absurd ( has no isolated points). Then and we have that so every singleton is nowhere dense and is of first category because it is a countable union of nowhere dense sets.
2) Suppose is a , that is, such that every is open. As is dense, then each is dense, because . But then and
which implies that is of first category. Then is of first category. Absurd, because is complete.
|Title||no countable dense subset of a complete metric space is a|
|Date of creation||2013-03-22 14:59:06|
|Last modified on||2013-03-22 14:59:06|
|Last modified by||gumau (3545)|