lecture 23

inputs/lecture_23.tex

@@ -161,7 +161,12 @@ $F(z_k, x_1) \to  0$, $F(z_k, x_2) \to 0$.
     So $(F_\alpha)_{\alpha \le \beta}$ is a strictly
     increasing chain of closed subsets.
     But $X$ is second countable,
-    so $\beta$ is countable.
+    so $\beta$ is countable:
+    Let $\{U_n\} = \cB$ be a countable basis
+    and for $\alpha$ let $U_\alpha \in \cB$
+    be such that $U_\alpha \cap F_\alpha = \emptyset$
+    and $U_\alpha \cap F_{\alpha+1} \neq \emptyset$.
+    Then $\alpha \mapsto U_\alpha$ is an injection.
 \end{proof}