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Josia Pietsch 2023-10-19 17:08:56 +02:00
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inputs/lecture_01.tex
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@ -42,14 +42,14 @@ Recall the following notions:
and $U_i \subsetneq X_i$ for only finitely many $i$. and $U_i \subsetneq X_i$ for only finitely many $i$.
\end{definition} \end{definition}
\begin{fact} \begin{fact}
Countable products of separable spaces are separable, Countable products of separable spaces are separable.
\end{fact} \end{fact}
\begin{definition} \begin{definition}
A topological space $X$ is \vocab{second countable}, A topological space $X$ is \vocab{second countable},
if it has a countable base. if it has a countable base.
\end{definition} \end{definition}
If $X$ is a topological space. Let $X$ be a topological space.
Then if $X$ is second countable, it is also separable. If $X$ is second countable, it is also separable.
However the converse of this does not hold. However the converse of this does not hold.
\begin{example} \begin{example}
@ -81,7 +81,8 @@ However the converse of this does not hold.
For a metric space, the following are equivalent: For a metric space, the following are equivalent:
\begin{itemize} \begin{itemize}
\item compact, \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} \item complete and \vocab{totally bounded}
(for all $\epsilon > 0$ we can cover the space with finitely many $\epsilon$-balls). (for all $\epsilon > 0$ we can cover the space with finitely many $\epsilon$-balls).
\end{itemize} \end{itemize}