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.

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