Show that the irrationals are a Baire space.
Solution
Let \(\mathbb{P} = \mathbb{R} \setminus \mathbb{Q}\) with the subspace topology.
1. Representation of \(\mathbb{P}\) as a \(G_\delta\) subset of \(\mathbb{R}\):Proof:
1.1. Enumerate the countable set of rational numbers as \(\mathbb{Q} = \{q_n\}_{n=1}^\infty\).
1.2. For each \(n \ge 1\), the singleton \(\{q_n\}\) is closed in \(\mathbb{R}\), so \(U_n = \mathbb{R} \setminus \{q_n\}\) is an open and dense subset of \(\mathbb{R}\).
1.3. The subspace of irrational numbers is:\[\mathbb{P} = \mathbb{R} \setminus \mathbb{Q} = \bigcap_{n=1}^\infty U_n.\]
2. Verification of the dense open intersection condition:Proof:
2.1. Let \(\{V_k\}_{k=1}^\infty\) be a countable sequence of dense open subsets of the subspace \(\mathbb{P}\).
2.2. Since each \(V_k\) is open in \(\mathbb{P}\), there exists an open set \(W_k \subseteq \mathbb{R}\) such that \(V_k = W_k \cap \mathbb{P}\).
2.3. Because \(V_k\) is dense in \(\mathbb{P}\) and \(\mathbb{P}\) is dense in \(\mathbb{R}\), the open set \(W_k\) is dense in \(\mathbb{R}\) for each \(k \ge 1\).
2.4. The combined collection \(\{W_k\}_{k=1}^\infty \cup \{U_n\}_{n=1}^\infty\) is a countable family of open and dense subsets of \(\mathbb{R}\).
2.5. By the Baire category theorem, the complete metric space \(\mathbb{R}\) is a Baire space, so the countable intersection:\[\bigcap_{k=1}^\infty W_k \cap \bigcap_{n=1}^\infty U_n = \left(\bigcap_{k=1}^\infty W_k\right) \cap \mathbb{P} = \bigcap_{k=1}^\infty V_k\]is dense in \(\mathbb{R}\).
2.6. Since this intersection lies in \(\mathbb{P}\) and is dense in \(\mathbb{R}\), it is dense in the subspace \(\mathbb{P}\).
3. Q.E.D.By step 2, every countable intersection of dense open subsets of \(\mathbb{P}\) is dense in \(\mathbb{P}\).