Progress over the past week.

src/topology/main/baire.tex

@@ -6,10 +6,10 @@
 \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.
+        \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}