small changes

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Josia Pietsch 2023-10-19 17:08:56 +02:00
parent a7c3cd0d3b
commit 1713fec189
Signed by: josia
GPG Key ID: E70B571D66986A2D
1 changed files with 5 additions and 4 deletions

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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}