lecture 4
This commit is contained in:
parent
1713fec189
commit
3bb3c4e75d
@ -107,6 +107,7 @@
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}
|
||||
\label{thm:cantortopolish}
|
||||
Let $X \neq \emptyset$
|
||||
be a perfect Polish space.
|
||||
Then there is an embedding
|
||||
@ -216,7 +217,7 @@
|
||||
\item $F_\emptyset = X$,
|
||||
\item $F_s$ is $F_\sigma$ for all $s$.
|
||||
\item The $F_{s \concat i}$ partition $F_s$,
|
||||
i.e.~$F_{s} = \bigsqcup_i F_{s \concat i}$.
|
||||
i.e.~$F_{s} = \bigsqcup_i F_{s \concat i}$. % TODO change notation?
|
||||
|
||||
Furthermore we want that
|
||||
$\overline{F_{s \concat i}} \subseteq F_s$
|
||||
|
156
inputs/lecture_04.tex
Normal file
156
inputs/lecture_04.tex
Normal file
@ -0,0 +1,156 @@
|
||||
\lecture{04}{2023-10-20}{}
|
||||
\begin{remark}
|
||||
Some of $F_s$ might be empty.
|
||||
\end{remark}
|
||||
|
||||
\begin{refproof}{thm:bairetopolish}
|
||||
Take
|
||||
\[D = \{x \in \cN : \bigcap_{n} F_{x\defon{n}} \neq \emptyset\}.\]
|
||||
|
||||
Since $\ldots \supseteq F_{x\defon{n}} \supseteq \overline{F_{x\defon{n+1}}} \supseteq F_{x\defon{n+1}} \supseteq \ldots$
|
||||
we have
|
||||
\[
|
||||
\bigcap_{n} F_{x\defon{n}} = \bigcap_{n} \overline{F_{x\defon{n}}}.
|
||||
\]
|
||||
|
||||
$f\colon D \to X$ is determined by
|
||||
\[
|
||||
\{f(x)\} = \bigcap_{n} F_{x\defon{n}}
|
||||
\]
|
||||
|
||||
$f$ is injective and continuous.
|
||||
The proof of this is exactly the same as in
|
||||
\yaref{thm:cantortopolish}.
|
||||
|
||||
\begin{claim}
|
||||
\label{thm:bairetopolish:c1}
|
||||
$D$ is closed.
|
||||
\end{claim}
|
||||
\begin{refproof}{thm:bairetopolish:c1}
|
||||
Let $x_n$ be a series in $D$
|
||||
converging to $x$ in $\cN$.
|
||||
Then $x \in \cN$.
|
||||
\begin{claim}
|
||||
$(f(x_n))$ is Cauchy.
|
||||
\end{claim}
|
||||
\begin{subproof}
|
||||
Let $\epsilon > 0$.
|
||||
Take $N$ such that $\diam(F_{x\defon{n}}) < \epsilon$.
|
||||
Take $M$ such that for all $m \ge M$,
|
||||
$x_m\defon{N} = x\defon{N}$.
|
||||
Then for all $m, n \ge M$,
|
||||
we have that $f(x_m), f(x_n) \in F_{x\defon{N}}$.
|
||||
So $d(f(x_m), f(x_n)) < \epsilon$
|
||||
we have that $(f(x_n))$ is Cauchy.
|
||||
|
||||
Since $(X,d)$ is complete,
|
||||
there exists $y = \lim_n f(x_n)$.
|
||||
|
||||
Since for all $m \ge M$, $f(x_m) \in F_{x\defon{N}}$,
|
||||
we get that $y \in \overline{F_{x\defon{N}}}$.
|
||||
|
||||
Note that for $N' > N$ by the same argument
|
||||
we get $y \in \overline{F_{x\defon{N'}}}$.
|
||||
Hence
|
||||
\[y \in \bigcap_{n} \overline{F_{x\defon{n}}} = \bigcap_{n} F_{x\defon{n}},\]
|
||||
i.e.~$y \in D$ and $y = f(x)$.
|
||||
\end{subproof}
|
||||
\end{refproof}
|
||||
|
||||
We extend $f$ to $g\colon\cN \to X$
|
||||
in the following way:
|
||||
|
||||
Take $S \coloneqq \{s \in \N^{<\N}: \exists x \in D, n \in \N.~x=s\defon{n}\}$.
|
||||
Clearly $S$ is a pruned tree.
|
||||
Moreover, since $D$ is closed, we have that\todo{Proof this (homework?)}
|
||||
\[
|
||||
D = [S] = \{x \in \N^\N : \forall n \in \N.~x\defon{n} \in S\}.
|
||||
\]
|
||||
We construct a \vocab{retraction} $r\colon\cN \to D$
|
||||
(i.e.~$r = \id$ on $D$ and $r$ is a continuous surjection).
|
||||
Then $g \coloneqq f \circ r$.
|
||||
|
||||
To construct $r$, we will define by induction
|
||||
$\phi\colon \N^{<\N} \to S$ by induction on the length
|
||||
such that
|
||||
\begin{itemize}
|
||||
\item $s \subseteq t \implies \phi(s) \subseteq \phi(t)$,
|
||||
\item $|s| = \phi(|s|)$,
|
||||
\item if $s \in S$, then $\phi(s) = s$.
|
||||
\end{itemize}
|
||||
Let $\phi(\emptyset) = \emptyset$.
|
||||
Suppose that $\phi(t)$ is defined.
|
||||
If $t\concat a \in S$, then set
|
||||
$\phi(t\concat a) \coloneqq t\concat a$.
|
||||
Otherwise take some $b$ such that
|
||||
$t\concat b \in S$ and define
|
||||
$\phi(t\concat a) \coloneqq \phi(t)\concat b$.
|
||||
This is possible since $S$ is pruned.
|
||||
|
||||
Let $r\colon \cN = [\N^{<\N}] \to [S] = D$
|
||||
be the function defined by $r(x) = \bigcup_n f(x\defon{n})$.
|
||||
|
||||
$r$ is continuous, since
|
||||
$d_{\cN}(r(x), r(y)) \le d_{\cN}(x,y)$. % Lipschitz
|
||||
It is immediate that $r$ is a retraction.
|
||||
\end{refproof}
|
||||
|
||||
\subsection{Meager and Comeager Sets}
|
||||
|
||||
\begin{definition}
|
||||
Let $X$ be a topological space, $A \subseteq X$.
|
||||
We say that $A$ is \vocab{nowhere dense} (\vocab{nwd}),
|
||||
if $\inter(\overline{A}) = \emptyset$.
|
||||
Equivalently
|
||||
\begin{itemize}
|
||||
\item $\overline{A}$ is nwd,
|
||||
\item $X \setminus A$ is dense in $X$,
|
||||
\item $\forall \emptyset \neq U \overset{\text{open}}{\subseteq} X.~
|
||||
\exists \emptyset \neq V \overset{\text{open}}{\subseteq} U.~
|
||||
V\cap A = \emptyset$.
|
||||
(If we intersect $A$ with an open $U$,
|
||||
then $A \cap U$ is not dense in $U$).
|
||||
\end{itemize}
|
||||
\todo{Think about this}
|
||||
|
||||
A set $B \subseteq X$ is \vocab{meager}
|
||||
(or \vocab{first category}),
|
||||
iff it is a countable union of nwd sets.
|
||||
|
||||
The complement of a meager set is called
|
||||
\vocab{comeager}.
|
||||
|
||||
\end{definition}
|
||||
\begin{example}
|
||||
$\Q \subseteq \R$ is meager.
|
||||
\end{example}
|
||||
\begin{notation}
|
||||
Let $A, B \subseteq X$.
|
||||
We write $A =^\ast B$
|
||||
iff the \vocab{symmetric difference},
|
||||
$A \mathop{\triangle} B \coloneqq (A\setminus B) \cup (B \setminus A)$,
|
||||
is meager.
|
||||
\end{notation}
|
||||
\begin{remark}
|
||||
$=^\ast$ is an equivalence relation.
|
||||
\end{remark}
|
||||
\begin{definition}
|
||||
A set $A \subseteq X$
|
||||
has the \vocab{Baire property} (\vocab{BP})
|
||||
if $A =^\ast U$ for some $U \overset{\text{open}}{\subseteq} X$.
|
||||
\end{definition}
|
||||
Note that open sets and meager sets have the Baire property.
|
||||
|
||||
|
||||
|
||||
|
||||
% \begin{example}
|
||||
% $\Q \subseteq \R$ is $F_\sigma$.
|
||||
%
|
||||
% $\R \setminus \Q \subseteq \R$ is $G_\delta$.
|
||||
%
|
||||
% $\Q \subseteq \R$ is not $G_{\delta}$.
|
||||
% (It is dense and meager,
|
||||
% hence it can not be $G_\delta$,
|
||||
% by the Baire category theorem).
|
||||
% \end{example}
|
@ -48,4 +48,5 @@
|
||||
\RequirePackage{mkessler-mathfig}
|
||||
\RequirePackage{mkessler-unicodechar}
|
||||
\RequirePackage{mkessler-mathfixes} % Load this last since it renews behaviour
|
||||
\DeclareMathOperator{\inter}{int} % interior
|
||||
\newcommand{\defon}[1]{|_{#1}} % TODO
|
||||
|
@ -27,6 +27,7 @@
|
||||
\input{inputs/lecture_01}
|
||||
\input{inputs/lecture_02}
|
||||
\input{inputs/lecture_03}
|
||||
\input{inputs/lecture_04}
|
||||
|
||||
|
||||
|
||||
|
Loading…
Reference in New Issue
Block a user