Merge branch 'main' of https://git.jrpie.de/jrpie-notes/logic-3-abstract-topological-dynamics-and-descriptive-set-theory

inputs/lecture_01.tex

@@ -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).
 

inputs/lecture_02.tex

@@ -96,7 +96,7 @@
      \end{IEEEeqnarray*}
 
      As for an open $U$, $f_Y$ is an embedding.
-     Since $X \times \R^{\N}$ 
+     Since $X \times \R^{\N}$
      is completely metrizable,
      so is the closed set $f_Y(Y) \subseteq X \times \R^\N$.
 
@@ -122,7 +122,7 @@
          $U_1$ (open in $Y)$ and $U_2$ (open in $X$ ) of $x$,
          such that $\diam_{d_Y}(U) < \frac{1}{n}$
          and $U_2 \cap Y = U_1$.
-         Additionally choose $x \in U_3$ open in $X$ 
+         Additionally choose $x \in U_3$ open in $X$
          with $\diam_{d}(U_3) < \frac{1}{n}$.
          Then consider $U_2 \cap U_3 \subseteq V_n$.
          Hence $Y \subseteq  \bigcap_{n \in \N}  V_n$.
@@ -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}