Determine whether or not \(\mathbb{R}_\ell\) is a Baire space.
Solution
\(\mathbb{R}_\ell\) is a Baire space. Let \(G_1, G_2, \ldots\) be dense open subsets of \(\mathbb{R}_\ell\) and let \(U_0\) be a nonempty open subset. It suffices to show that \(U_0 \cap \bigcap_n G_n \neq \varnothing\).
1. There are real numbers \(a_n < b_n\), \(n \ge 0\), with \([a_0, b_0] \subseteq U_0\) and \([a_{n+1}, b_{n+1}] \subseteq G_{n+1} \cap [a_n, b_n)\) for \(n \ge 0\).
The sets \([a, b)\) form a basis of \(\mathbb{R}_\ell\). The nonempty open set \(U_0\) contains some \([a_0, b_0')\); put \(b_0 = (a_0 + b_0')/2\). Given \(a_n < b_n\), the set \(G_{n+1} \cap [a_n, b_n)\) is open, and nonempty because \(G_{n+1}\) is dense. It contains some \([a_{n+1}, b')\) with \(a_{n+1} < b'\); put \(b_{n+1} = (a_{n+1} + b')/2\). Then \([a_{n+1}, b_{n+1}] \subseteq [a_{n+1}, b') \subseteq G_{n+1} \cap [a_n, b_n)\).
2. Q.E.D.
By step 1 the closed bounded intervals \([a_n, b_n]\) of \(\RR\) are nested and nonempty, so they have a common point \(x\) by compactness. Then \(x \in [a_0, b_0] \subseteq U_0\) and \(x \in [a_{n}, b_{n}] \subseteq G_{n}\) for every \(n \ge 1\).