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Diffstat (limited to 'library/topology/urysohn2.tex')
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1 files changed, 172 insertions, 11 deletions
diff --git a/library/topology/urysohn2.tex b/library/topology/urysohn2.tex index ea49a6c..c0b46c4 100644 --- a/library/topology/urysohn2.tex +++ b/library/topology/urysohn2.tex @@ -15,6 +15,34 @@ \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 $\dom{X} \subseteq \naturals$. +\end{definition} \begin{abbreviation}\label{urysohnspace} @@ -26,13 +54,81 @@ \end{abbreviation} -\begin{definition}\label{intervalclosed} - $\intervalclosed{a}{b} = \{x \in \reals \mid a \leq x \leq b\}$. +\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 $\at{X}{n} \subseteq \carrier[Y]$ and for all $m \in \dom{X}$ such that $m > n$ we have $\at{X}{n} \subseteq \at{X}{m}$. +\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 $A,B \in \closeds{X}$. @@ -42,21 +138,86 @@ and $f(A) = \zero$ and $f(B)= 1$ and $f$ is continuous. \end{theorem} \begin{proof} - - - Define $f : X \to \reals$ such that $f(x) = $ + 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} - &(x + k) &\text{if} x \in X \land k \in \naturals - & x &\text{if} x \neq \zero - & \zero & \text{if} x = \zero - % & x ,x \in X <- will result in technicly ambigus parse + &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. + We have $1, \zero \in N$. + We show that $U_0$ is a chain of subsets in $X$. + \begin{subproof} + We have $\dom{U_0} \subseteq \naturals$. + We have for all $n \in \dom{U_0}$ we have $\at{U_0}{n} \subseteq \carrier[X]$ by \cref{topological_prebasis_iff_covering_family,union_as_unions,union_absorb_subseteq_left,subset_transitive,setminus_subseteq}. + We have $\dom{U_0} = \{\zero, 1\}$. + + It suffices to show that for all $n \in \dom{U_0}$ we have for all $m \in \dom{U_0}$ such that $m > n$ we have $\at{U_0}{n} \subseteq \at{U_0}{m}$. + + Fix $n \in \dom{U_0}$. + Fix $m \in \dom{U_0}$. + + \begin{byCase} + \caseOf{$n \neq \zero$.} + Trivial. + \caseOf{$n = \zero$.} + \begin{byCase} + \caseOf{$m = \zero$.} + Trivial. + \caseOf{$m \neq \zero$.} + We have $A \subseteq A'$. + We have $\at{U_0}{\zero} = A$ by assumption. + We have $\at{U_0}{1}= A'$ by assumption. + \end{byCase} + \end{byCase} + + \end{subproof} + $U_0$ is a urysohnchain of $X$. + + %We are done with the first step, now we want to prove that we have U a sequence of these chain with U_0 the first chain. + + - Trivial. - \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 +% + |