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@ 42,14 +42,14 @@ Recall the following notions:


and $U_i \subsetneq X_i$ for only finitely many $i$.


\end{definition}


\begin{fact}


Countable products of separable spaces are separable,


Countable products of separable spaces are separable.


\end{fact}


\begin{definition}


A topological space $X$ is \vocab{second countable},


if it has a countable base.


\end{definition}


If $X$ is a topological space.


Then if $X$ is second countable, it is also separable.


Let $X$ be a topological space.


If $X$ is second countable, it is also separable.


However the converse of this does not hold.




\begin{example}



@ 81,7 +81,8 @@ However the converse of this does not hold.


For a metric space, the following are equivalent:


\begin{itemize}


\item compact,


\item \vocab{sequentially compact} (every sequence has a convergent subsequence),


\item \vocab{sequentially compact}


(every sequence has a convergent subsequence),


\item complete and \vocab{totally bounded}


(for all $\epsilon > 0$ we can cover the space with finitely many $\epsilon$balls).


\end{itemize}




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