Tagged Questions
2
votes
2answers
129 views
Dense subset of Cantor set homeomorphic to the Baire space
Does anyone know a proof that the Cantor set, $\{0,1\}^{\mathbb{N}}$, has a dense subset homeomorphic to the Baire space, $\mathbb{N}^{\mathbb{N}}$? Thank you.
3
votes
1answer
105 views
Question missing condition in Royden Exercise 7.42 b, about Baire Category
In Royden's Real Analysis P164 Q7.42b,
It assumes that $X$ and $Y$ are complete metric spaces. Let $O$ be a dense open set in $X \times Y$.
Assertion: Then there is a $G \subset X$ which is a ...
34
votes
5answers
1k views
Why is it that $\mathbb{Q}$ cannot be homeomorphic to _any_ complete metric space?
Why is it that $\mathbb{Q}$ cannot be homeomorphic to any complete metric space?
Certainly $\mathbb{Q}$ is not a complete metric space. But completeness is not a topological invariant, so why is the ...
