I need to prove that the euclidean line is a Baire Space.

I have this definitions for Baire Space:

  • A space $Y$ is a Baire space if the intersection of each countable family of open dense sets in $Y$ is dense.

-$X$ is a Baire Space if the following condition holds: Given any countable collection $\{A_n\}$ of closed sets of $X$ each of which has empty interior in $X$, their union $\cup A_n$ also has empty interior in $X$.

I know that $E^1_u$ with the usual topology is $\mathbb{R}$ and this is of second category and we have that Baire space is of second category but "Not all space of the second category is Baire Space". So I have troubless to do this proof. Thank you very much.


We can prove that the space $\Bbb R$ is Baire using the fact that each its non-empty open subset has an open homeomorphic copy of $\Bbb R$ which is of second category in itself, or directly as it the following theorem from “Baire spaces” by Haworth and McCoy.

enter image description here


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.