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\head{NOTE 158}\endhead A proof that form 341 implies form 10.
Form 341 states: Every Lindel\"of metric space is second countable,
and [10 A] is: $UT(\aleph_{0},< \aleph_{0},\aleph_{0})$: The
union of denumerably many pairwise disjoint finite sets is
denumerable. Let $A = \{A_i: i\in\omega\}$, where each $A_i$
is finite and the $A_i$'s are pairwise disjoint. Let $X$ be the one
point compactification of $\bigcup A$ with the discrete topology.
The space $X$ is Lindel\"of, so by 341, $X$ is second countable.
This implies that $\bigcup A$ is countable, which proves form 10.
\bye