diff options
Diffstat (limited to 'library/topology')
| -rw-r--r-- | library/topology/real-topological-space.tex | 2 | ||||
| -rw-r--r-- | library/topology/urysohn.tex | 140 | ||||
| -rw-r--r-- | library/topology/urysohn2.tex | 254 |
3 files changed, 267 insertions, 129 deletions
diff --git a/library/topology/real-topological-space.tex b/library/topology/real-topological-space.tex index b2e5ea9..c76fd46 100644 --- a/library/topology/real-topological-space.tex +++ b/library/topology/real-topological-space.tex @@ -11,7 +11,7 @@ \import{function.tex} -\section{The canonical topology on $\mathbb{R}$} +\section{Topology Reals} \begin{definition}\label{topological_basis_reals_eps_ball} $\topoBasisReals = \{ \epsBall{x}{\epsilon} \mid x \in \reals, \epsilon \in \realsplus\}$. diff --git a/library/topology/urysohn.tex b/library/topology/urysohn.tex index ff6a231..ae03273 100644 --- a/library/topology/urysohn.tex +++ b/library/topology/urysohn.tex @@ -36,7 +36,7 @@ The first tept will be a formalisation of chain constructions. % $\overline{A_{i-1}} \subset \interior{A_{i}}$. % In this case we call the chain legal. -\begin{definition}\label{one_to_n_set} +\begin{definition}\label{urysohnone_one_to_n_set} $\seq{m}{n} = \{x \in \naturals \mid m \leq x \leq n\}$. \end{definition} @@ -48,7 +48,7 @@ The first tept will be a formalisation of chain constructions. % together with the existence of an indexing function. % %%----------------------- -\begin{struct}\label{sequence} +\begin{struct}\label{urysohnone_sequence} A sequence $X$ is a onesorted structure equipped with \begin{enumerate} \item $\indexx$ @@ -57,8 +57,8 @@ The first tept will be a formalisation of chain constructions. \end{enumerate} such that \begin{enumerate} - \item\label{indexset_is_subset_naturals} $\indexxset[X] \subseteq \naturals$. - \item\label{index_is_bijection} $\indexx[X]$ is a bijection from $\indexxset[X]$ to $\carrier[X]$. + \item\label{urysohnone_indexset_is_subset_naturals} $\indexxset[X] \subseteq \naturals$. + \item\label{urysohnone_index_is_bijection} $\indexx[X]$ is a bijection from $\indexxset[X]$ to $\carrier[X]$. \end{enumerate} \end{struct} @@ -67,12 +67,12 @@ The first tept will be a formalisation of chain constructions. -\begin{definition}\label{cahin_of_subsets} +\begin{definition}\label{urysohnone_cahin_of_subsets} $C$ is a chain of subsets iff $C$ is a sequence and for all $n,m \in \indexxset[C]$ such that $n < m$ we have $\indexx[C](n) \subseteq \indexx[C](m)$. \end{definition} -\begin{definition}\label{chain_of_n_subsets} +\begin{definition}\label{urysohnone_chain_of_n_subsets} $C$ is a chain of $n$ subsets iff $C$ is a chain of subsets and $n \in \indexxset[C]$ and for all $m \in \naturals$ such that $m \leq n$ we have $m \in \indexxset[C]$. @@ -84,7 +84,7 @@ The first tept will be a formalisation of chain constructions. % and also for the subproof of continuity of the limit. -% \begin{definition}\label{legal_chain} +% \begin{definition}\label{urysohnone_legal_chain} % $C$ is a legal chain of subsets of $X$ iff % $C \subseteq \pow{X}$. %and % %there exist $f \in \funs{C}{\naturals}$ such that @@ -106,49 +106,49 @@ The first tept will be a formalisation of chain constructions. \subsection{staircase function} -\begin{definition}\label{minimum} +\begin{definition}\label{urysohnone_minimum} $\min{X} = \{x \in X \mid \forall y \in X. x \leq y \}$. \end{definition} -\begin{definition}\label{maximum} +\begin{definition}\label{urysohnone_maximum} $\max{X} = \{x \in X \mid \forall y \in X. x \geq y \}$. \end{definition} -\begin{definition}\label{intervalclosed} +\begin{definition}\label{urysohnone_intervalclosed} $\intervalclosed{a}{b} = \{x \in \reals \mid a \leq x \leq b\}$. \end{definition} -\begin{definition}\label{intervalopen} +\begin{definition}\label{urysohnone_intervalopen} $\intervalopen{a}{b} = \{ x \in \reals \mid a < x < b\}$. \end{definition} -\begin{struct}\label{staircase_function} +\begin{struct}\label{urysohnone_staircase_function} A staircase function $f$ is a onesorted structure equipped with \begin{enumerate} \item $\chain$ \end{enumerate} such that \begin{enumerate} - \item \label{staircase_is_function} $f$ is a function to $\intervalclosed{\zero}{1}$. - \item \label{staircase_domain} $\dom{f}$ is a topological space. - \item \label{staricase_def_chain} $C$ is a chain of subsets. - \item \label{staircase_chain_is_in_domain} for all $x \in C$ we have $x \subseteq \dom{f}$. - \item \label{staircase_behavoir_index_zero} $f(\indexx[C](1))= 1$. - \item \label{staircase_behavoir_index_n} $f(\dom{f}\setminus \unions{C}) = \zero$. - \item \label{staircase_chain_indeset} There exist $n$ such that $\indexxset[C] = \seq{\zero}{n}$. - \item \label{staircase_behavoir_index_arbetrray} for all $n \in \indexxset[C]$ + \item \label{urysohnone_staircase_is_function} $f$ is a function to $\intervalclosed{\zero}{1}$. + \item \label{urysohnone_staircase_domain} $\dom{f}$ is a topological space. + \item \label{urysohnone_staricase_def_chain} $C$ is a chain of subsets. + \item \label{urysohnone_staircase_chain_is_in_domain} for all $x \in C$ we have $x \subseteq \dom{f}$. + \item \label{urysohnone_staircase_behavoir_index_zero} $f(\indexx[C](1))= 1$. + \item \label{urysohnone_staircase_behavoir_index_n} $f(\dom{f}\setminus \unions{C}) = \zero$. + \item \label{urysohnone_staircase_chain_indeset} There exist $n$ such that $\indexxset[C] = \seq{\zero}{n}$. + \item \label{urysohnone_staircase_behavoir_index_arbetrray} for all $n \in \indexxset[C]$ such that $n \neq \zero$ we have $f(\indexx[C](n) \setminus \indexx[C](n-1)) = \rfrac{n}{ \max{\indexxset[C]} }$. \end{enumerate} \end{struct} -\begin{definition}\label{legal_staircase} +\begin{definition}\label{urysohnone_legal_staircase} $f$ is a legal staircase function iff $f$ is a staircase function and for all $n,m \in \indexxset[\chain[f]]$ such that $n \leq m$ we have $f(\indexx[\chain[f]](n)) \leq f(\indexx[\chain[f]](m))$. \end{definition} -\begin{abbreviation}\label{urysohnspace} +\begin{abbreviation}\label{urysohnone_urysohnspace} $X$ is a urysohn space iff $X$ is a topological space and for all $A,B \in \closeds{X}$ such that $A \inter B = \emptyset$ @@ -156,49 +156,49 @@ The first tept will be a formalisation of chain constructions. such that $A \subseteq A'$ and $B \subseteq B'$ and $A' \inter B' = \emptyset$. \end{abbreviation} -\begin{definition}\label{urysohnchain} +\begin{definition}\label{urysohnone_urysohnchain} $C$ is a urysohnchain in $X$ of cardinality $k$ iff %<---- TODO: cardinality unused! $C$ is a chain of subsets and for all $A \in C$ we have $A \subseteq X$ and for all $n,m \in \indexxset[C]$ such that $n < m$ we have $\closure{\indexx[C](n)}{X} \subseteq \interior{\indexx[C](m)}{X}$. \end{definition} -\begin{definition}\label{urysohnchain_without_cardinality} +\begin{definition}\label{urysohnone_urysohnchain_without_cardinality} $C$ is a urysohnchain in $X$ iff $C$ is a chain of subsets and for all $A \in C$ we have $A \subseteq X$ and for all $n,m \in \indexxset[C]$ such that $n < m$ we have $\closure{\indexx[C](n)}{X} \subseteq \interior{\indexx[C](m)}{X}$. \end{definition} -\begin{abbreviation}\label{infinte_sequence} +\begin{abbreviation}\label{urysohnone_infinte_sequence} $S$ is a infinite sequence iff $S$ is a sequence and $\indexxset[S]$ is infinite. \end{abbreviation} -\begin{definition}\label{infinite_product} +\begin{definition}\label{urysohnone_infinite_product} $X$ is the infinite product of $Y$ iff $X$ is a infinite sequence and for all $i \in \indexxset[X]$ we have $\indexx[X](i) = Y$. \end{definition} -\begin{definition}\label{refinmant} +\begin{definition}\label{urysohnone_refinmant} $C'$ is a refinmant of $C$ iff $C'$ is a urysohnchain in $X$ and for all $x \in C$ we have $x \in C'$ and for all $y \in C$ such that $y \subset x$ we have there exist $c \in C'$ such that $y \subset c \subset x$ and for all $g \in C'$ such that $g \neq c$ we have not $y \subset g \subset x$. \end{definition} -\begin{abbreviation}\label{two} +\begin{abbreviation}\label{urysohnone_two} $\two = \suc{1}$. \end{abbreviation} -\begin{lemma}\label{two_in_reals} +\begin{lemma}\label{urysohnone_two_in_reals} $\two \in \reals$. \end{lemma} -\begin{lemma}\label{two_in_naturals} +\begin{lemma}\label{urysohnone_two_in_naturals} $\two \in \naturals$. \end{lemma} -\begin{inductive}\label{power_of_two} +\begin{inductive}\label{urysohnone_power_of_two} Define $\powerOfTwoSet \subseteq (\naturals \times \naturals)$. \begin{enumerate} \item $(\zero, 1) \in \powerOfTwoSet$. @@ -206,45 +206,45 @@ The first tept will be a formalisation of chain constructions. \end{enumerate} \end{inductive} -\begin{abbreviation}\label{pot} +\begin{abbreviation}\label{urysohnone_pot} $\pot = \powerOfTwoSet$. \end{abbreviation} -\begin{lemma}\label{dom_pot} +\begin{lemma}\label{urysohnone_dom_pot} $\dom{\pot} = \naturals$. \end{lemma} \begin{proof} Omitted. \end{proof} -\begin{lemma}\label{ran_pot} +\begin{lemma}\label{urysohnone_ran_pot} $\ran{\pot} \subseteq \naturals$. \end{lemma} -\begin{axiom}\label{pot1} +\begin{axiom}\label{urysohnone_pot1} $\pot \in \funs{\naturals}{\naturals}$. \end{axiom} -\begin{axiom}\label{pot2} +\begin{axiom}\label{urysohnone_pot2} For all $n \in \naturals$ we have there exist $k\in \naturals$ such that $(n, k) \in \powerOfTwoSet$ and $\apply{\pot}{n}=k$. %$\pot(n) = k$ iff there exist $x \in \powerOfTwoSet$ such that $x = (n,k)$. \end{axiom} %Without this abbreviation \pot cant be sed as a function in the standard sense -\begin{abbreviation}\label{pot_as_function} +\begin{abbreviation}\label{urysohnone_pot_as_function} $\pot(n) = \apply{\pot}{n}$. \end{abbreviation} %Take all points, besids one but then take all open sets not containing x but all other, so \{x\} has to be closed -\begin{axiom}\label{hausdorff_implies_singltons_closed} +\begin{axiom}\label{urysohnone_hausdorff_implies_singltons_closed} For all $X$ such that $X$ is Hausdorff we have for all $x \in \carrier[X]$ we have $\{x\}$ is closed in $X$. \end{axiom} -\begin{lemma}\label{urysohn_set_in_between} +\begin{lemma}\label{urysohnone_urysohn_set_in_between} Let $X$ be a urysohn space. Suppose $A,B \in \closeds{X}$. Suppose $A \subset B$. @@ -284,7 +284,7 @@ The first tept will be a formalisation of chain constructions. \end{proof} -\begin{proposition}\label{urysohnchain_induction_begin} +\begin{proposition}\label{urysohnone_urysohnchain_induction_begin} Let $X$ be a urysohn space. Suppose $A,B \in \closeds{X}$. Suppose $A \inter B$ is empty. @@ -347,7 +347,7 @@ The first tept will be a formalisation of chain constructions. \end{proof} -\begin{proposition}\label{urysohnchain_induction_begin_step_two} +\begin{proposition}\label{urysohnone_urysohnchain_induction_begin_step_two} Let $X$ be a urysohn space. Suppose $A,B \in \closeds{X}$. Suppose $A \inter B$ is empty. @@ -364,7 +364,7 @@ The first tept will be a formalisation of chain constructions. -\begin{proposition}\label{t_four_propositon} +\begin{proposition}\label{urysohnone_t_four_propositon} Let $X$ be a urysohn space. Then for all $A,B \subseteq X$ such that $\closure{A}{X} \subseteq \interior{B}{X}$ we have there exists $C \subseteq X$ such that @@ -376,7 +376,7 @@ The first tept will be a formalisation of chain constructions. -\begin{proposition}\label{urysohnchain_induction_step_existence} +\begin{proposition}\label{urysohnone_urysohnchain_induction_step_existence} Let $X$ be a urysohn space. Suppose $U$ is a urysohnchain in $X$. Then there exist $U'$ such that $U'$ is a refinmant of $U$ and $U'$ is a urysohnchain in $X$. @@ -390,7 +390,7 @@ The first tept will be a formalisation of chain constructions. % such that $\closure{\indexx[U](n)}{X} \subseteq \interior{C}{X} \subseteq \closure{C}{X} \subseteq \interior{\indexx[U](n+1)}{X}$. - %\begin{definition}\label{refinmant} + %\begin{definition}\label{urysohnone_refinmant} % $C'$ is a refinmant of $C$ iff for all $x \in C$ we have $x \in C'$ and % for all $y \in C$ such that $y \subset x$ % we have there exist $c \in C'$ such that $y \subset c \subset x$ @@ -404,7 +404,7 @@ The first tept will be a formalisation of chain constructions. -\begin{proposition}\label{existence_of_staircase_function} +\begin{proposition}\label{urysohnone_existence_of_staircase_function} Let $X$ be a urysohn space. Suppose $U$ is a urysohnchain in $X$ and $U$ has cardinality $k$. Suppose $k \neq \zero$. @@ -416,7 +416,7 @@ The first tept will be a formalisation of chain constructions. Omitted. \end{proof} -\begin{abbreviation}\label{refinment_abbreviation} +\begin{abbreviation}\label{urysohnone_refinment_abbreviation} $x \refine y$ iff $x$ is a refinmant of $y$. \end{abbreviation} @@ -424,27 +424,27 @@ The first tept will be a formalisation of chain constructions. -\begin{abbreviation}\label{sequence_of_functions} +\begin{abbreviation}\label{urysohnone_sequence_of_functions} $f$ is a sequence of functions iff $f$ is a sequence and for all $g \in \carrier[f]$ we have $g$ is a function. \end{abbreviation} -\begin{abbreviation}\label{sequence_in_reals} +\begin{abbreviation}\label{urysohnone_sequence_in_reals} $s$ is a sequence of real numbers iff $s$ is a sequence and for all $r \in \carrier[s]$ we have $r \in \reals$. \end{abbreviation} -\begin{axiom}\label{abs_behavior1} +\begin{axiom}\label{urysohnone_abs_behavior1} If $x \geq \zero$ then $\abs{x} = x$. \end{axiom} -\begin{axiom}\label{abs_behavior2} +\begin{axiom}\label{urysohnone_abs_behavior2} If $x < \zero$ then $\abs{x} = \neg{x}$. \end{axiom} -\begin{abbreviation}\label{converge} +\begin{abbreviation}\label{urysohnone_converge} $s$ converges iff $s$ is a sequence of real numbers and $\indexxset[s]$ is infinite and for all $\epsilon \in \reals$ such that $\epsilon > \zero$ we have @@ -454,7 +454,7 @@ The first tept will be a formalisation of chain constructions. \end{abbreviation} -\begin{definition}\label{limit_of_sequence} +\begin{definition}\label{urysohnone_limit_of_sequence} $x$ is the limit of $s$ iff $s$ is a sequence of real numbers and $x \in \reals$ and for all $\epsilon \in \reals$ such that $\epsilon > \zero$ @@ -463,7 +463,7 @@ The first tept will be a formalisation of chain constructions. we have $\abs{x - \indexx[s](n)} < \epsilon$. \end{definition} -\begin{proposition}\label{existence_of_limit} +\begin{proposition}\label{urysohnone_existence_of_limit} Let $s$ be a sequence of real numbers. Then $s$ converges iff there exist $x \in \reals$ such that $x$ is the limit of $s$. @@ -472,22 +472,22 @@ The first tept will be a formalisation of chain constructions. Omitted. \end{proof} -\begin{definition}\label{limit_sequence} +\begin{definition}\label{urysohnone_limit_sequence} $x$ is the limit sequence of $f$ iff $x$ is a sequence and $\indexxset[x] = \dom{f}$ and for all $n \in \indexxset[x]$ we have $\indexx[x](n) = f(n)$. \end{definition} -\begin{definition}\label{realsminus} +\begin{definition}\label{urysohnone_realsminus} $\realsminus = \{r \in \reals \mid r < \zero\}$. \end{definition} -\begin{abbreviation}\label{realsplus} +\begin{abbreviation}\label{urysohnone_realsplus} $\realsplus = \reals \setminus \realsminus$. \end{abbreviation} -\begin{proposition}\label{intervalclosed_subseteq_reals} +\begin{proposition}\label{urysohnone_intervalclosed_subseteq_reals} Suppose $a,b \in \reals$. Suppose $a < b$. Then $\intervalclosed{a}{b} \subseteq \reals$. @@ -495,7 +495,7 @@ The first tept will be a formalisation of chain constructions. -\begin{lemma}\label{fraction1} +\begin{lemma}\label{urysohnone_fraction1} Let $x \in \reals$. Then for all $y \in \reals$ such that $x \neq y$ we have there exists $r \in \rationals$ such that $y < r < x$ or $x < r < y$. \end{lemma} @@ -503,7 +503,7 @@ The first tept will be a formalisation of chain constructions. Omitted. \end{proof} -\begin{lemma}\label{frection2} +\begin{lemma}\label{urysohnone_frection2} Suppose $a,b \in \reals$. Suppose $a < b$. Then $\intervalopen{a}{b}$ is infinite. @@ -512,7 +512,7 @@ The first tept will be a formalisation of chain constructions. Omitted. \end{proof} -\begin{lemma}\label{frection3} +\begin{lemma}\label{urysohnone_frection3} Suppose $a \in \reals$. Suppose $a < \zero$. Then there exist $N \in \naturals$ such that for all $N' \in \naturals$ such that $N' > N$ we have $\zero < \rfrac{1}{\pot(N')} < a$. @@ -521,7 +521,7 @@ The first tept will be a formalisation of chain constructions. Omitted. \end{proof} -\begin{proposition}\label{fraction4} +\begin{proposition}\label{urysohnone_fraction4} Suppose $a,b,\epsilon \in \reals$. Suppose $\epsilon > \zero$. $\abs{a - b} < \epsilon$ iff $b \in \intervalopen{(a - \epsilon)}{(a + \epsilon)}$. @@ -530,7 +530,7 @@ The first tept will be a formalisation of chain constructions. Omitted. \end{proof} -\begin{proposition}\label{fraction5} +\begin{proposition}\label{urysohnone_fraction5} Suppose $a,b,\epsilon \in \reals$. Suppose $\epsilon > \zero$. $b \in \intervalopen{(a - \epsilon)}{(a + \epsilon)}$ iff $a \in \intervalopen{(b - \epsilon)}{(b + \epsilon)}$. @@ -539,17 +539,17 @@ The first tept will be a formalisation of chain constructions. Omitted. \end{proof} -\begin{proposition}\label{fraction6} +\begin{proposition}\label{urysohnone_fraction6} Suppose $a,\epsilon \in \reals$. Suppose $\epsilon > \zero$. $\intervalopen{(a - \epsilon)}{(a + \epsilon)} = \{r \in \reals \mid (a - \epsilon) < r < (a + \epsilon)\} $. \end{proposition} -\begin{abbreviation}\label{epsilonball} +\begin{abbreviation}\label{urysohnone_epsilonball} $\epsBall{a}{\epsilon} = \intervalopen{(a - \epsilon)}{(a + \epsilon)}$. \end{abbreviation} -\begin{proposition}\label{fraction7} +\begin{proposition}\label{urysohnone_fraction7} Suppose $a,\epsilon \in \reals$. Suppose $\epsilon > \zero$. Then there exist $b \in \rationals$ such that $b \in \epsBall{a}{\epsilon}$. @@ -561,11 +561,11 @@ The first tept will be a formalisation of chain constructions. -%\begin{definition}\label{sequencetwo} +%\begin{definition}\label{urysohnone_sequencetwo} % $Z$ is a sequencetwo iff $Z = (N,f,B)$ and $N \subseteq \naturals$ and $f$ is a bijection from $N$ to $B$. %\end{definition} % -%\begin{proposition}\label{sequence_existence} +%\begin{proposition}\label{urysohnone_sequence_existence} % Suppose $N \subseteq \naturals$. % Suppose $M \subseteq \naturals$. % Suppose $N = M$. @@ -586,7 +586,7 @@ The first tept will be a formalisation of chain constructions. -\begin{theorem}\label{urysohn} +\begin{theorem}\label{urysohnone_urysohn1} Let $X$ be a urysohn space. Suppose $A,B \in \closeds{X}$. Suppose $A \inter B$ is empty. @@ -599,7 +599,7 @@ The first tept will be a formalisation of chain constructions. There exist $\eta$ such that $\carrier[\eta] = \{A, (\carrier[X] \setminus B)\}$ and $\indexxset[\eta] = \{\zero, 1\}$ and $\indexx[\eta](\zero) = A$ - and $\indexx[\eta](1) = (\carrier[X] \setminus B)$ by \cref{urysohnchain_induction_begin}. + and $\indexx[\eta](1) = (\carrier[X] \setminus B)$ by \cref{urysohnone_urysohnchain_induction_begin}. We show that there exist $\zeta$ such that $\zeta$ is a sequence and $\indexxset[\zeta] = \naturals$ @@ -919,6 +919,6 @@ The first tept will be a formalisation of chain constructions. % \end{subproof} \end{proof} % -%\begin{theorem}\label{safe} +%\begin{theorem}\label{urysohnone_safe} % Contradiction. %\end{theorem} diff --git a/library/topology/urysohn2.tex b/library/topology/urysohn2.tex index ce6d742..08841da 100644 --- a/library/topology/urysohn2.tex +++ b/library/topology/urysohn2.tex @@ -40,7 +40,7 @@ \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 $\at{X}{n} \subseteq \at{X}{m}$. + $X$ 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 $n < m$ we have $\at{X}{n} \subseteq \at{X}{m}$. \end{definition} @@ -49,11 +49,11 @@ \end{definition} \begin{definition}\label{urysohn_finer_set} - $A$ is finer between $X$ to $Y$ in $U$ iff $\closure{X}{U} \subseteq \interior{A}{U}$ and $\closure{A}{U} \subseteq \interior{Y}{U}$. + $A$ is finer between $B$ to $C$ in $X$ iff $\closure{B}{X} \subseteq \interior{A}{X}$ and $\closure{A}{X} \subseteq \interior{C}{X}$. \end{definition} \begin{definition}\label{finer} %<-- verfeinerung - $Y$ is finer then $X$ 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 $\at{Y}{k}$ is finer between $\at{X}{n}$ to $\at{X}{m}$ in $U$. + $A$ is finer then $B$ in $X$ iff for all $n \in \dom{B}$ we have $\at{B}{n} \in \ran{A}$ and for all $m \in \dom{B}$ such that $n < m$ we have there exist $k \in \dom{A}$ such that $\at{A}{k}$ is finer between $\at{B}{n}$ to $\at{B}{m}$ in $X$. \end{definition} \begin{definition}\label{follower_index} @@ -92,6 +92,46 @@ $f$ is consistent on $X$ to $Y$ iff $f$ is a bijection from $\dom{X}$ to $\dom{Y}$ and for all $n,m \in \dom{X}$ such that $n < m$ we have $f(n) < f(m)$. \end{definition} + +%\begin{definition}\label{staircase} +% $f$ is a staircase function adapted to $U$ in $X$ iff $U$ is a urysohnchain of $X$ and $f$ is a function from $\carrier[X]$ to $\reals$ and there exist $k \in \naturals$ such that $k = \max{\dom{U}}$ and for all $x,y \in \carrier[X]$ such that $y \in \carrier[X] \setminus \at{U}{k}$ and $x \in \at{U}{k}$ we have $f(y) = 1$ and there exist $n,m \in \dom{U}$ such that $n$ follows $m$ in $\dom{U}$ and $x \in (\at{U}{m} \setminus \at{U}{n})$ and $f(x)= \rfrac{m}{k}$. +%\end{definition} + + +\begin{definition}\label{staircase_step_value1} + $a$ is the staircase step value at $y$ of $U$ in $X$ iff there exist $n,m \in \dom{U}$ such that $n$ follows $m$ in $\dom{U}$ and $y \in \closure{\at{U}{n}}{X} \setminus \closure{\at{U}{m}}{X}$ and $a = \rfrac{n}{\max{\dom{U}}}$. +\end{definition} + +\begin{definition}\label{staircase_step_value2} + $a$ is the staircase step valuetwo at $y$ of $U$ in $X$ iff either if $y \in (\carrier[X] \setminus \closure{\at{U}{\max{\dom{U}}}}{X})$ then $a = 1$ or $a$ is the staircase step valuethree at $y$ of $U$ in $X$. +\end{definition} + +\begin{definition}\label{staircase_step_value3} + $a$ is the staircase step valuethree at $y$ of $U$ in $X$ iff if $y \in \closure{\at{U}{\min{\dom{U}}}}{X}$ then $f(z) = \zero$. +\end{definition} + + +\begin{definition}\label{staircase2} + $f$ is a staircase function adapted to $U$ in $X$ iff $U$ is a urysohnchain of $X$ and $f$ is a function from $\carrier[X]$ to $\reals$ and for all $y \in \carrier[X]$ we have either $f(y)$ is the staircase step value at $y$ of $U$ in $X$ or $f(y)$ is the staircase step valuetwo at $y$ of $U$ in $X$. +\end{definition} + +\begin{definition}\label{staircase_sequence} + $S$ is staircase sequence of $U$ in $X$ iff $S$ is a sequence and $U$ is a lifted urysohnchain of $X$ and $\dom{U} = \dom{S}$ and for all $n \in \dom{U}$ we have $\at{S}{n}$ is a staircase function adapted to $\at{U}{n}$ in $X$. +\end{definition} + +\begin{definition}\label{staircase_limit_point} + $x$ is the staircase limit of $S$ with $y$ iff $x \in \reals$ and for all $\epsilon \in \realsplus$ there exist $n_0 \in \naturals$ such that for all $n \in \naturals$ such that $n_0 \rless n$ we have $\apply{\at{S}{n}}{y} \in \epsBall{x}{\epsilon}$. +\end{definition} + +%\begin{definition}\label{staircase_limit_function} +% $f$ is a limit function of a staircase $S$ iff $S$ is staircase sequence of $U$ and $U$ is a lifted urysohnchain of $X$ and $\dom{f} = \carrier[X]$ and for all $x \in \dom{f}$ we have $f(x)$ is the staircase limit of $S$ with $x$ and $f$ is a function from $\carrier[X]$ to $\reals$. +%\end{definition} +% +\begin{definition}\label{staircase_limit_function} + $f$ is the limit function of staircase $S$ together with $U$ and $X$ iff $S$ is staircase sequence of $U$ in $X$ and $U$ is a lifted urysohnchain of $X$ and $\dom{f} = \carrier[X]$ and for all $x \in \dom{f}$ we have $f(x)$ is the staircase limit of $S$ with $x$ and $f$ is a function from $\carrier[X]$ to $\reals$. +\end{definition} + + \begin{proposition}\label{naturals_in_transitive} $\naturals$ is a \in-transitive set. \end{proposition} @@ -659,33 +699,26 @@ \end{proof} -\begin{definition}\label{staircase} - $f$ is a staircase function adapted to $U$ in $X$ iff $U$ is a urysohnchain of $X$ and for all $x,n,m,k$ such that $k = \max{\dom{U}}$ and $n,m \in \dom{U}$ and $n$ follows $m$ in $\dom{U}$ and $x \in (\at{U}{m} \setminus \at{U}{n})$ we have $f(x)= \rfrac{m}{k}$. -\end{definition} - -\begin{definition}\label{staircase_sequence} - $S$ is staircase sequence of $U$ iff $S$ is a sequence and $U$ is a lifted urysohnchain of $X$ and $\dom{U} = \dom{S}$ and for all $n \in \dom{U}$ we have $\at{S}{n}$ is a staircase function adapted to $\at{U}{n}$ in $U$. -\end{definition} - -\begin{definition}\label{staircase_limit_point} - $x$ is the staircase limit of $S$ with $y$ iff $x \in \reals$ and for all $\epsilon \in \realsplus$ there exist $n_0 \in \naturals$ such that for all $n \in \naturals$ such that $n_0 \rless n$ we have $\apply{\at{S}{n}}{y} \in \epsBall{x}{\epsilon}$. -\end{definition} - -\begin{definition}\label{staircase_limit_function} - $f$ is a limit function of a staircase $S$ iff $S$ is staircase sequence of $U$ and $U$ is a lifted urysohnchain of $X$ and $\dom{f} = \carrier[X]$ and for all $x \in \dom{f}$ we have $f(x)$ is the staircase limit of $S$ with $x$ and $f$ is a function from $\carrier[X]$ to $\reals$. -\end{definition} +\begin{proposition}\label{staircase_ran_in_zero_to_one} + Let $X$ be a urysohn space. + Suppose $U$ is a urysohnchain of $X$. + Suppose $f$ is a staircase function adapted to $U$ in $X$. + Then $\ran{f} \subseteq \intervalclosed{\zero}{1}$. +\end{proposition} +\begin{proof} + Omitted. +\end{proof} -%\begin{definition}\label{staircase_limit_function} -% $f$ is a limit function of staircase $S$ iff $S$ is staircase sequence of $U$ and $U$ is a lifted urysohnchain of $X$ and $\dom{f} = \carrier[X]$ and for all $x \in \dom{f}$ we have $f(x)$ is the staircase limit of $S$ with $x$ and $f$ is a function from $\carrier[X]$ to $\reals$. -%\end{definition} -% -%\begin{proposition}\label{staircase_limit_is_continuous} -% Suppose $X$ is a urysohnspace. -% Suppose $U$ is a lifted urysohnchain of $X$. -% Suppose $S$ is staircase sequence of $U$. -% Suppose $f$ is the limit function of a staircase $S$. -% Then $f$ is continuous. -%\end{proposition} +\begin{proposition}\label{staircase_limit_is_continuous} + Let $X$ be a urysohn space. + Suppose $U$ is a lifted urysohnchain of $X$. + Suppose $S$ is staircase sequence of $U$ in $X$. + Suppose $f$ is the limit function of staircase $S$ together with $U$ and $X$. + Then $f$ is continuous. +\end{proposition} +\begin{proof} + Omitted. +\end{proof} \begin{theorem}\label{urysohnsetinbeetween} Let $X$ be a urysohn space. @@ -712,8 +745,26 @@ Omitted. \end{proof} +\begin{lemma}\label{fractions_between_zero_one} + Suppose $n,m \in \naturals$. + Suppose $m > n$. + Then $\zero \leq \rfrac{n}{m} \leq 1$. +\end{lemma} +\begin{proof} + Omitted. +\end{proof} +\begin{lemma}\label{intervalclosed_border_is_elem} + Suppose $a,b \in \reals$. + Suppose $a < b$. + Then $a,b \in \intervalclosed{a}{b}$. +\end{lemma} +\begin{lemma}\label{urysohnchain_subseteqrel} + Let $X$ be a urysohn space. + Suppose $U$ is a urysohnchain of $X$. + Then for all $n,m \in \dom{U}$ such that $n < m$ we have $\at{U}{n} \subseteq \at{U}{m}$. +\end{lemma} \begin{theorem}\label{urysohn} @@ -721,8 +772,8 @@ 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. + There exist $f$ such that $f \in \funs{\carrier[X]}{\intervalclosed{\zero}{1}}$ and $f$ is continuous + and for all $a,b$ such that $a \in A$ and $b \in B$ we have $f(a)= \zero$ and $f(b) = 1$. \end{theorem} \begin{proof} Let $X' = \carrier[X]$. @@ -796,46 +847,133 @@ \end{subproof} Take $U$ such that $U$ is a lifted urysohnchain of $X$ and $\at{U}{\zero} = U_0$. - We show that there exist $S$ such that $S$ is staircase sequence of $U$. + We show that there exist $S$ such that $S$ is staircase sequence of $U$ in $X$. \begin{subproof} Omitted. \end{subproof} - Take $S$ such that $S$ is staircase sequence of $U$. + Take $S$ such that $S$ is staircase sequence of $U$ in $X$. %For all $x \in \carrier[X]$ we have there exist $r,R$ such that $r \in \reals$ and $R$ is a sequence of reals and $\dom{R} = \naturals$ and $R$ converge to $r$ and for all $n \in \naturals$ we have $\at{R}{n} = \apply{\at{S}{n}}{x}$. % %We show that for all $x \in \carrier[X]$ there exists $r \in \intervalclosed{a}{b}$ such that for . -% + We show that there exist $f$ such that $f$ is the limit function of staircase $S$ together with $U$ and $X$. + \begin{subproof} + Omitted. + \end{subproof} + Take $f$ such that $f$ is the limit function of staircase $S$ together with $U$ and $X$. + Then $f$ is continuous. + We show that $\dom{f} = \carrier[X]$. + \begin{subproof} + Trivial. + \end{subproof} + $f$ is a function. + We show that $\ran{f} \subseteq \intervalclosed{\zero}{1}$. + \begin{subproof} + It suffices to show that $f$ is a function to $\intervalclosed{\zero}{1}$. + It suffices to show that for all $x \in \dom{f}$ we have $f(x) \in \intervalclosed{\zero}{1}$. + Fix $x \in \dom{f}$. + $f(x)$ is the staircase limit of $S$ with $x$. + Therefore $f(x) \in \reals$. + + We show that for all $n \in \naturals$ we have $\apply{\at{S}{n}}{x} \in \intervalclosed{\zero}{1}$. + \begin{subproof} + Fix $n \in \naturals$. + Let $g = \at{S}{n}$. + Let $U' = \at{U}{n}$. + $\at{S}{n}$ is a staircase function adapted to $\at{U}{n}$ in $X$. + $g$ is a staircase function adapted to $U'$ in $X$. + $U'$ is a urysohnchain of $X$. + $g$ is a function from $\carrier[X]$ to $\reals$. + It suffices to show that $\ran{g} \subseteq \intervalclosed{\zero}{1}$ by \cref{function_apply_default,reals_axiom_zero_in_reals,intervalclosed,one_is_positiv,function_apply_elim,inter,inter_absorb_supseteq_left,ran_iff,funs_is_relation,funs_is_function,staircase2}. + It suffices to show that for all $x \in \dom{g}$ we have $g(x) \in \intervalclosed{\zero}{1}$. + Fix $x\in \dom{g}$. + Then $x \in \carrier[X]$. + \begin{byCase} + \caseOf{$x \in (\carrier[X] \setminus \closure{\at{U'}{\max{\dom{U'}}}}{X})$.} + Therefore $x \notin \closure{\at{U'}{\max{\dom{U'}}}}{X}$. + Therefore $x \notin \closure{\at{U'}{\min{\dom{U'}}}}{X}$. + Therefore $x \notin (\closure{\at{U'}{\max{\dom{U'}}}}{X}\setminus \closure{\at{U'}{\min{\dom{U'}}}}{X})$. + Then $g(x) = 1$ . + \caseOf{$x \in \closure{\at{U'}{\max{\dom{U'}}}}{X}$.} + \begin{byCase} + \caseOf{$x \in \closure{\at{U'}{\min{\dom{U'}}}}{X}$.} + $g(x) = \zero$. + \caseOf{$x \in (\closure{\at{U'}{\max{\dom{U'}}}}{X}\setminus \closure{\at{U'}{\min{\dom{U'}}}}{X})$.} + Then $g(x)$ is the staircase step value at $x$ of $U'$ in $X$. + \end{byCase} + \end{byCase} + + + + %$\at{S}{n}$ is a staircase function adapted to $\at{U}{n}$ in $X$. + %$\at{U}{n}$ is a urysohnchain of $X$. + %$\at{S}{n}$ is a function from $\carrier[X]$ to $\reals$. + %there exist $k \in \naturals$ such that $k = \max{\dom{\at{U}{n}}}$. + %Take $k \in \naturals$ such that $k = \max{\dom{\at{U}{n}}}$. + %\begin{byCase} + % \caseOf{$x \in \carrier[X] \setminus \at{\at{U}{n}}{k}$.} + % $1 \in \intervalclosed{\zero}{1}$. + % We show that for all $y \in (\carrier[X] \setminus \at{\at{U}{n}}{k})$ we have $\apply{\at{S}{n}}{y} = 1$. + % \begin{subproof} + % Omitted. + % \end{subproof} + % Then $\apply{\at{S}{n}}{x} = 1$. + % \caseOf{$x \notin \carrier[X] \setminus \at{\at{U}{n}}{k}$.} + % %There exist $n',m' \in \dom{\at{U}{n}}$ such that $n'$ follows $m'$ in $\dom{\at{U}{n}}$ and $x \in (\at{\at{U}{n}}{n'} \setminus \at{\at{U}{n}}{m'})$. + % Take $n',m' \in \dom{\at{U}{n}}$ such that $n'$ follows $m'$ in $\dom{\at{U}{n}}$ and $x \in (\at{\at{U}{n}}{n'} \setminus \at{\at{U}{n}}{m'})$. + % Then $\apply{\at{S}{n}}{x} = \rfrac{m'}{k'}$. + % It suffices to show that $\rfrac{m'}{k'} \in \intervalclosed{\zero}{1}$. + % $\zero \leq m' \leq k$. + %\end{byCase} + %%It suffices to show that $\zero \leq \apply{\at{S}{n}}{x} \leq 1$. + %%It suffices to show that $\ran{\at{S}{n}} \subseteq \intervalclosed{\zero}{1}$. + \end{subproof} + + Suppose not. + Then $f(x) < \zero$ or $f(x) > 1$ by \cref{reals_order_total,reals_axiom_zero_in_reals,intervalclosed,one_is_positiv,one_in_reals}. + For all $\epsilon \in \realsplus$ we have there exist $m \in \naturals$ such that $\apply{\at{S}{m}}{x} \in \epsBall{f(x)}{\epsilon}$ by \cref{plus_one_order,naturals_is_equal_to_two_times_naturals,subseteq,naturals_subseteq_reals,staircase_limit_point}. + \begin{byCase} + \caseOf{$f(x) < \zero$.} + Let $\delta = \zero - f(x)$. + $\delta \in \realsplus$. + It suffices to show that for all $n \in \naturals$ we have $\apply{\at{S}{n}}{x} \notin \epsBall{f(x)}{\delta}$. + Fix $n \in \naturals$. + $\at{S}{n}$ is a staircase function adapted to $\at{U}{n}$ in $X$. + For all $y \in \epsBall{f(x)}{\delta}$ we have $y < \zero$ by \cref{epsilon_ball,minus_behavior1,minus_behavior3,minus,apply,intervalopen}. + It suffices to show that $\apply{\at{S}{n}}{x} \in \intervalclosed{\zero}{1}$. + Trivial. + \caseOf{$f(x) > 1$.} + Let $\delta = f(x) - 1$. + $\delta \in \realsplus$. + It suffices to show that for all $n \in \naturals$ we have $\apply{\at{S}{n}}{x} \notin \epsBall{f(x)}{\delta}$. + Fix $n \in \naturals$. + $\at{S}{n}$ is a staircase function adapted to $\at{U}{n}$ in $X$. + For all $y \in \epsBall{f(x)}{\delta}$ we have $y > 1$ by \cref{epsilon_ball,reals_addition_minus_behavior2,minus_in_reals,apply,reals_addition_minus_behavior1,minus,reals_add,realsplus_in_reals,one_in_reals,reals_axiom_kommu,intervalopen}. + It suffices to show that $\apply{\at{S}{n}}{x} \in \intervalclosed{\zero}{1}$. + Trivial. + \end{byCase} + + \end{subproof} + Therefore $f \in \funs{\carrier[X]}{\intervalclosed{\zero}{1}}$ by \cref{staircase_limit_function,surj_to_fun,fun_to_surj,neq_witness,inters_of_ordinals_elem,times_tuple_elim,img_singleton_iff,foundation,subseteq_emptyset_iff,inter_eq_left_implies_subseteq,inter_emptyset,funs_intro,fun_ran_iff,not_in_subseteq}. + + We show that for all $a \in A$ we have $f(a) = \zero$. + \begin{subproof} + Omitted. + \end{subproof} + We show that for all $b \in B$ we have $f(b) = 1$. + \begin{subproof} + Omitted. + \end{subproof} + - \end{proof} -\begin{theorem}\label{safe} - Contradiction. -\end{theorem} +%\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 -% |