lecture 23
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Josia Pietsch 2024-01-19 11:56:29 +01:00
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@ -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 So $(F_\alpha)_{\alpha \le \beta}$ is a strictly
increasing chain of closed subsets. increasing chain of closed subsets.
But $X$ is second countable, 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} \end{proof}