countable clopen base
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Josia Pietsch 2023-12-06 13:57:23 +01:00
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@ -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)