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src/topology/main/baire.tex

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+\section{Baire Spaces}
+\label{section:baire}
+
+
+\begin{definition}[Baire Space]
+\label{definition:baire}
+    Let $X$ be a topological space, then the following are equivalent:
+    \begin{enumerate}
+        \item For any $\seq{A_n}$ nowhere dense, $\bigcup_{n \in \nat}A_n \subsetneq X$.
+        \item For any $\seq{A_n}$ closed with empty interior, $\bigcup_{n \in \nat}A_n \subsetneq X$.
+        \item For any $\seq{A_n}$ closed with $\bigcup_{n \in \nat}A_n = X$, there exists $N \in \nat$ such that $\bigcup_{n \le N}A_n$ has non-empty interior.
+        \item For any $\seq{U_n}$ open and dense, $\bigcap_{n \in \nat}U_n$ is dense.
+    \end{enumerate}
+    If the above holds, then $X$ is a \textbf{Baire space}.
+\end{definition}
+
+\begin{theorem}[Baire Category Theorem]
+\label{theorem:baire}
+    Let $X$ be a topological space, then the following are sufficient conditions for $X$ to be Baire:
+    \begin{enumerate}
+        \item $X$ is completely metrisable.
+        \item $X$ is locally compact.
+    \end{enumerate}
+\end{theorem}