Josia Pietsch
c986475c77
Some checks failed
Build latex and deploy / checkout (push) Failing after 1m4s
77 lines
2.2 KiB
TeX
77 lines
2.2 KiB
TeX
\tutorial{01}{2023-10-17}{}
|
|
|
|
|
|
\begin{fact}
|
|
A countable product of separable spaces $(X_n)_{n \in \N}$ is separable.
|
|
\end{fact}
|
|
\begin{proof}
|
|
Choose a countable dense subset $D_n \subseteq X_n$
|
|
Fix some point $(a_1,a_2,\ldots) \in \prod_n X_n$
|
|
and consider $\bigcup_{i \in \N} \prod_{n \le i} D_n \times \prod_{n > i} \{a_n\}$.
|
|
\end{proof}
|
|
|
|
\begin{fact}
|
|
\begin{itemize}
|
|
\item Let $X$ be a topological space.
|
|
Then $X$ \nth{2} countable $\implies$ X separable.
|
|
\item If $X$ is a metric space and separable,
|
|
then $X$ is \nth{2} countable.
|
|
\end{itemize}
|
|
\end{fact}
|
|
\begin{proof}
|
|
For the first point, choose some point from every basic open set.
|
|
|
|
For the second point consider balls of rational radius
|
|
around the points of a countable dense subset.
|
|
\end{proof}
|
|
|
|
\begin{definition}
|
|
A topological space is \vocab{Lindelöf}
|
|
iff every open cover has a countable subcover.
|
|
\end{definition}
|
|
|
|
|
|
\begin{fact}
|
|
Let $X$ be a metric space.
|
|
If $X$ is Lindelöf,
|
|
then it is \nth{2} countable.
|
|
\end{fact}
|
|
\begin{proof}
|
|
For all $q \in \Q$
|
|
consider the cover $B_q(x), x \in X$
|
|
and choose a countable subcover.
|
|
The union of these subcovers is
|
|
a countable base.
|
|
\end{proof}
|
|
\begin{fact}
|
|
Let $X$ be a topological space.
|
|
If $X$ is \nth{2} countable,
|
|
then it is Lindelöff.
|
|
\end{fact}
|
|
\begin{proof}
|
|
Let $A_0, A_1,\ldots$
|
|
be a countable base.
|
|
|
|
Let $\{U_i\}_{i \in I}$
|
|
be a cover.
|
|
Consider $J \coloneqq \{j : \exists i \in I.~A_j \in U_i\}$.
|
|
For every $j \in J$ choose a $U_i$ such that
|
|
$A_j \subseteq U_j$.
|
|
Let $I' \subseteq I$ be the subset of chosen indices.
|
|
Then $\{U_i\}_{i \in I'}$ is a countable subcover.
|
|
\end{proof}
|
|
\begin{remark}
|
|
For metric spaces the notions
|
|
of being \nth{2} countable, separable
|
|
and Lindelöf coincide.
|
|
|
|
In arbitrary topological spaces,
|
|
Lindelöf is the weakest of these notions.
|
|
\end{remark}
|
|
|
|
\begin{definition}+
|
|
A metric space $X$ is \vocab{totally bounded}
|
|
iff for every $\epsilon > 0$ there exists
|
|
a finite set of points $x_1,\ldots,x_n$
|
|
such that $X = \bigcup_{i=1}^n B_{\epsilon}(x_i)$.
|
|
\end{definition}
|