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\import{topology/topological-space.tex}
\import{topology/separation.tex}
\import{topology/continuous.tex}
\import{topology/basis.tex}
\import{numbers.tex}
\import{function.tex}
\import{set.tex}
\import{cardinal.tex}
\import{relation.tex}
\import{relation/uniqueness.tex}
\import{set/cons.tex}
\import{set/powerset.tex}
\import{set/fixpoint.tex}
\import{set/product.tex}
\section{Urysohns Lemma}
\begin{definition}\label{minimum}
$\min{X} = \{x \in X \mid \forall y \in X. x \leq y \}$.
\end{definition}
\begin{definition}\label{maximum}
$\max{X} = \{x \in X \mid \forall y \in X. x \geq y \}$.
\end{definition}
\begin{definition}\label{intervalclosed}
$\intervalclosed{a}{b} = \{x \in \reals \mid a \leq x \leq b\}$.
\end{definition}
\begin{definition}\label{intervalopen}
$\intervalopen{a}{b} = \{ x \in \reals \mid a < x < b\}$.
\end{definition}
\begin{definition}\label{one_to_n_set}
$\seq{m}{n} = \{x \in \naturals \mid m \leq x \leq n\}$.
\end{definition}
\begin{definition}\label{sequence}
$X$ is a sequence iff $X$ is a function and $\dom{f} \subseteq \naturals$.
\end{definition}
\begin{abbreviation}\label{urysohnspace}
$X$ is a urysohn space iff
$X$ is a topological space and
for all $A,B \in \closeds{X}$ such that $A \inter B = \emptyset$
we have there exist $A',B' \in \opens[X]$
such that $A \subseteq A'$ and $B \subseteq B'$ and $A' \inter B' = \emptyset$.
\end{abbreviation}
\begin{abbreviation}\label{at}
$\at{f}{n} = f(n)$.
\end{abbreviation}
\begin{definition}\label{chain_of_subsets}
$X$ is a chain of subsets in $Y$ iff $X$ is a sequence and for all $n \in \dom{X}$ we have $\at{X}{n} \subseteq \carrier[Y]$.
\end{definition}
\begin{definition}\label{urysohnchain}%<-- zulässig
$X$ is a urysohnchain of $Y$ iff $X$ is a chain of subsets in $Y$ and for all $n,m \in \dom{X}$ such that $n < m$ we have $\closure{\at{X}{n}}{Y} \subseteq \interior{\at{X}{m}}{Y}$.
\end{definition}
\begin{definition}\label{finer} %<-- verfeinerung
$X$ is finer then $Y$ in $U$ iff for all $n \in \dom{X}$ we have $\at{X}{n} \in \ran{Y}$ and for all $m \in \dom{X}$ such that $n < m$ we have there exist $k \in \dom{Y}$ such that $ \closure{\at{X}{n}}{U} \subseteq \interior{\at{Y}{k}}{U} \subseteq \closure{\at{Y}{k}}{U} \subseteq \interior{\at{X}{m}}{U}$.
\end{definition}
\begin{definition}\label{sequence_of_reals}
$X$ is a sequence of reals iff $\ran{X} \subseteq \reals$.
\end{definition}
\begin{axiom}\label{abs_behavior1}
If $x \geq \zero$ then $\abs{x} = x$.
\end{axiom}
\begin{axiom}\label{abs_behavior2}
If $x < \zero$ then $\abs{x} = \neg{x}$.
\end{axiom}
\begin{definition}\label{realsminus}
$\realsminus = \{r \in \reals \mid r < \zero\}$.
\end{definition}
\begin{definition}\label{realsplus}
$\realsplus = \reals \setminus \realsminus$.
\end{definition}
\begin{definition}\label{epsilon_ball}
$\epsBall{x}{\epsilon} = \intervalopen{x-\epsilon}{x+\epsilon}$.
\end{definition}
\begin{definition}\label{pointwise_convergence}
$X$ converge to $x$ iff for all $\epsilon \in \realsplus$ there exist $N \in \dom{X}$ such that for all $n \in \dom{X}$ such that $n > N$ we have $\at{X}{n} \in \epsBall{x}{\epsilon}$.
\end{definition}
\begin{proposition}\label{iff_sequence}
Suppose $X$ is a function.
Suppose $\dom{X} \subseteq \naturals$.
Then $X$ is a sequence.
\end{proposition}
\begin{theorem}\label{urysohnsetinbeetween}
Let $X$ be a urysohn space.
Suppose $A,B \in \closeds{X}$.
Suppose $\closure{A}{X} \subseteq \interior{B}{X}$.
Suppose $\carrier[X]$ is inhabited.
There exist $U \subseteq \carrier[X]$ such that $U$ is closed in $X$ and $\closure{A}{X} \subseteq \interior{U}{X} \subseteq \closure{U}{X} \subseteq \interior{B}{X}$.
\end{theorem}
\begin{proof}
Omitted.
\end{proof}
\begin{theorem}\label{urysohn}
Let $X$ be a urysohn space.
Suppose $A,B \in \closeds{X}$.
Suppose $A \inter B$ is empty.
Suppose $\carrier[X]$ is inhabited.
There exist $f$ such that $f \in \funs{\carrier[X]}{\intervalclosed{\zero}{1}}$
and $f(A) = \zero$ and $f(B)= 1$ and $f$ is continuous.
\end{theorem}
\begin{proof}
Let $X' = \carrier[X]$.
Let $N = \{\zero, 1\}$.
$1 = \suc{\zero}$.
$1 \in \naturals$ and $\zero \in \naturals$.
$N \subseteq \naturals$.
Let $A' = (X' \setminus B)$.
$B \subseteq X'$ by \cref{powerset_elim,closeds}.
$A \subseteq X'$.
Therefore $A \subseteq A'$.
Define $U_0: N \to \{A, A'\}$ such that $U_0(n) =$
\begin{cases}
&A &\text{if} n = \zero \\
&A' &\text{if} n = 1
\end{cases}
$U_0$ is a function.
$\dom{U_0} = N$.
$\dom{U_0} \subseteq \naturals$ by \cref{ran_converse}.
$U_0$ is a sequence.
$U_0$ is a chain of subsets in $X'$.
$U_0$ is a urysohnchain of $X$.
\end{proof}
\begin{theorem}\label{safe}
Contradiction.
\end{theorem}
%
%Ideen:
%Eine folge ist ein Funktion mit domain \subseteq Natürlichenzahlen. als predicat
%
%zulässig und verfeinerung von ketten als predicat definieren.
%
%limits und punkt konvergenz als prädikat.
%
%
%Vor dem Beweis vor dem eigentlichen Beweis.
%die abgeleiteten Funktionen
%
%\derivedstiarcasefunction on A
%
%abbreviation: \at{f}{n} = f_{n}
%
%
%TODO:
%Reals ist ein topologischer Raum
%
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