54-00-EverySecondCountableSpaceIsSeparable.tex

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\pmtitle{every second countable space is separable}
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\pmtype{Proof}
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\pmrelated{Separable}

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\begin{document}
\begin{thm} \cite{kelley}
  Every second countable space is separable.
\end{thm}


\begin{proof}
Let $X$ be a second countable space and let $\cal B$ be a countable base.
For every non-empty set $B$ in $\cal B$, choose a point $x_B\in B$. The set $A$ 
of all such points $x_B$ is clearly countable and it's also dense 
since any open set intersects it and thus the whole space is the closure of $A$.
That is, $A$ is a countably dense subset of $X$. Therefore, $X$ is separable.
\end{proof}

\begin{thebibliography}{9}
\bibitem{kelley}
J.L. Kelley, \emph{General Topology}, D. van Nostrand Company, Inc., 1955.
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