Josia Pietsch 2023-10-20 11:53:40 +02:00
commit f838245711
Signed by: josia
GPG key ID: E70B571D66986A2D
2 changed files with 14 additions and 10 deletions

View file

@ -28,7 +28,7 @@ i.e.~it is really a topological property.
\end{example}
Polish spaces behave very nicely.
We will see that uncountable polish spaces have size $2^{\aleph_0}$.
We will see that uncountable polish spaces have size $2^{\aleph_0}$. % TODO: mathfrak c for continuum
There are good notions of big (comeager)
and small (meager).

View file

@ -130,16 +130,20 @@
Now let $x \in \bigcap_{n \in \N} V_n$.
For each $n$ pick $x \in U_n \subseteq X$ open
satisfying (i), (ii), (iii).
W.l.o.g. the $U_n$ are decreasing.
From (i) and (ii) it follows that $x \in \overline{Y}$,
since we can consider a sequence of points $y_n \in U_n \cap Y$
and get $y_n \xrightarrow{d} x$.
On the other hand $\diam_{d_Y}(U_n \cap Y) \le \frac{1}{n}$,
so the $y_n$ form a Cauchy sequence with respect to $d_Y$,
since $\diam(U_n \cap Y) \xrightarrow{d_Y} 0$,
hence $\diam(\overline{U_n \cap Y}) \xrightarrow{d_Y} 0$.
$y_n$ converges to the unique point in $\bigcap_{n} \overline{U_n \cap Y}$.
Since the topologies agree, this point is $x$.
For all $n$ we have that $U_n' \coloneqq U_1 \cap \ldots \cap U_n$
is an open set containing $x$,
hence $U_n' \cap Y \neq \emptyset$.
Thus we may assume that the $U_i$ form a decreasing sequence.
We have that $\diam_{d_Y}(U_n \cap Y) \le \frac{1}{n}$.
If follows that the $y_n$ form a Cauchy sequence with respect to $d_Y$,
since $\diam(U_n \cap Y) \xrightarrow{d_Y} 0$
and thus $\diam(\overline{U_n \cap Y}) \xrightarrow{d_Y} 0$.
The sequence $y_n$ converges to the unique point in
$\bigcap_{n} \overline{U_n \cap Y}$.
Since the topologies agree, this point is $x$.
\end{refproof}
\end{refproof}