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