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Diffstat (limited to 'library/topology/urysohn2.tex')
| -rw-r--r-- | library/topology/urysohn2.tex | 157 |
1 files changed, 132 insertions, 25 deletions
diff --git a/library/topology/urysohn2.tex b/library/topology/urysohn2.tex index 71de210..e963951 100644 --- a/library/topology/urysohn2.tex +++ b/library/topology/urysohn2.tex @@ -15,23 +15,35 @@ \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{struct}\label{sequence} - A sequence $X$ is a onesorted structure equipped with - \begin{enumerate} - \item $\index$ - \item $\indexset$ - \end{enumerate} - such that - \begin{enumerate} - \item\label{indexset_is_subset_naturals} $\indexset[X] \subseteq \naturals$. - \item\label{index_is_bijection} $\index[X]$ is a bijection from $\indexset[X]$ to $\carrier[X]$. - \end{enumerate} -\end{struct} + +\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 @@ -42,10 +54,66 @@ \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 $\at{X}{n} \subseteq \carrier[Y]$. +\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{theorem}\label{urysohnsetinbeetween} Let $X$ be a urysohn space. Suppose $A,B \in \closeds{X}$. @@ -55,7 +123,6 @@ \end{theorem} - \begin{theorem}\label{urysohn} Let $X$ be a urysohn space. Suppose $A,B \in \closeds{X}$. @@ -76,8 +143,8 @@ Define $F : \eta \to \reals$ such that $F(x) =$ \begin{cases} & \zero &\text{if} x \in A\\ - & \rfrac{1}{1+1} &\text{if} x \in (\carrier[X] \setminus (A \union B))\\ - & 1 &\text{if} x \in B + & \rfrac{1}{1+1} &\text{if} x \in (\carrier[X] \setminus (A \union B))\\ + & 1 &\text{if} x \in B \end{cases} %Define $f : \naturals \to \pow{P}$ such that $f(x)=$ @@ -87,19 +154,59 @@ % & G & \text{if} x \in (\naturals \setminus \{1, \zero\}) \land G = \{g \in \pow{P} \mid g \in f(n-1) \lor (g \notin f(n-1) \land g \in P) \} %\end{cases} - Let $D = \{d \mid d \in \rationals \mid \zero \leq d \leq 1\}$. - Take $R$ such that for all $q \in D$ we have for all $S \in P$ we have $q \mathrel{R} S$ iff - $q = \zero \land S = A$ or $q = 1 \land S = H$ or - for all $q_1, q_2, S_1, S_2$ - such that $q_1 \leq q \leq q_2$ and $q_1 \mathrel{R} S_1$ and $q_2 \mathrel{R} S_2$ - we have $\closure{S_1}{X} \subseteq \interior{S}{X} \subseteq \closure{S}{X} \subseteq \interior{S_2}{X}$ - and $q \mathrel{R} S$. + %Let $D = \{d \mid d \in \rationals \mid \zero \leq d \leq 1\}$. + %Take $R$ such that for all $q \in D$ we have for all $S \in P$ we have $q \mathrel{R} S$ iff + % $q = \zero \land S = A$ or $q = 1 \land S = H$ or + % for all $q_1, q_2, S_1, S_2$ + % such that $q_1 \leq q \leq q_2$ and $q_1 \mathrel{R} S_1$ and $q_2 \mathrel{R} S_2$ + % we have $\closure{S_1}{X} \subseteq \interior{S}{X} \subseteq \closure{S}{X} \subseteq \interior{S_2}{X}$ + % and $q \mathrel{R} S$. - Trivial. + %Let $J = \{(n,f) \mid n denots the cardinality of a urysohn chain on which f is a staircase function\}$. +% + %Let $N = \naturals$. + %Define $j : N \to \funs{\carrier[X]}{\eals}$ such that $j(n) =$ + %\begin{cases} + % & f &\text{if} n \in N \land \exist w \in J. J=(n,f) + %\end{cases} + + + + + + + + + \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 +% + |