countable clopen base

inputs/tutorial_08.tex

@@ -110,7 +110,30 @@ for some $B_i \in \cB(Y_i)$.
 
 \nr 3
 \todo{Wait for mail}
-\todo{Find a countable clopen base}
+
+\begin{lemma}
+    Let $X$ be a second-countable topological space.
+    Then every base of $X$ contains a countable subset which
+    is also a base of $X$.
+\end{lemma}
+\begin{proof}
+    Let $\cC = \{C_n : n < \omega\}$
+    be a countable base of $X$
+    and let $\cB = \{B_i: i \in I\}$
+    be a base of $X$ with (possibly uncountable) index set $I$.
+
+    Fix $n < \omega$. It suffices to show that $C_n$ is a union
+    of countably many elements of $\cB$.
+    As  $\cB$ is a base, $C_n = \bigcup_{j \in J} B_j$
+    for some $J \subseteq I$.
+    Since $\cC$ is a base,
+    there exists $M_j \subseteq \N$ such that $B_j = \bigcup_{m \in M_j} C_m$
+    for all $j \in J$.
+    Let $M = \bigcup_{j \in J} M_j \subseteq \N$.
+    For each $m \in M$, there exists $f(m) \in J$
+    such that $m \in M_{f(m)}$.
+    Then $\bigcup_{m \in M} B_{f(m)} = C_n$.
+\end{proof}
 
 \begin{itemize}
     \item We use the same construction as in exercise 2 (a)