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Diffstat (limited to 'library/topology/urysohn.tex')
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1 files changed, 125 insertions, 8 deletions
diff --git a/library/topology/urysohn.tex b/library/topology/urysohn.tex index c795db8..bb28116 100644 --- a/library/topology/urysohn.tex +++ b/library/topology/urysohn.tex @@ -147,9 +147,9 @@ The first tept will be a formalisation of chain constructions. \end{definition} \begin{definition}\label{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 $y \subset c \subset x$ - and for all $g \in C$ such that $g \neq c$ we have not $y \subset g \subset x$. + $C'$ 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 $y \subset c \subset x$ + 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} @@ -165,14 +165,131 @@ The first tept will be a formalisation of chain constructions. \end{lemma} \begin{inductive}\label{power_of_two} - Define $\powerOfTwoSet \subseteq \naturals$. + Define $\powerOfTwoSet \subseteq (\naturals \times \naturals)$. \begin{enumerate} - \item $\two \in \powerOfTwoSet$. - \item If $n \in \powerOfTwoSet$, then $n \rmul \two \in \powerOfTwoSet$. + \item $(\zero, 1) \in \powerOfTwoSet$. + \item If $(m,k) \in \powerOfTwoSet$, then $(m+1, k \rmul \two) \in \powerOfTwoSet$. \end{enumerate} - \end{inductive} +\begin{abbreviation}\label{pot} + $\pot = \powerOfTwoSet$. +\end{abbreviation} + +\begin{lemma}\label{dom_pot} + $\dom{\pot} = \naturals$. +\end{lemma} +\begin{proof} + Omitted. +\end{proof} + +\begin{lemma}\label{ran_pot} + $\ran{\pot} \subseteq \naturals$. +\end{lemma} + + + + + + + + +\begin{proposition}\label{urysohnchain_induction_begin} + Let $X$ be a urysohn space. + Suppose $A,B \in \closeds{X}$. + Suppose $A \inter B$ is empty. + Suppose $U = \{A,(X \setminus B)\}$. + Suppose $\indexset[U]= \{\zero, 1\}$. + Suppose $\index[U](\zero) = A$. + Suppose $\index[U](1) = (X \setminus B)$. + Then $U$ is a urysohnchain in $X$. +\end{proposition} +\begin{proof} + We show that $U$ is a sequence. + \begin{subproof} + Omitted. + \end{subproof} + + We show that $A \subseteq (X \setminus B)$. + \begin{subproof} + Omitted. + \end{subproof} + + We show that $U$ is a chain of subsets. + \begin{subproof} + For all $n \in \indexset[U]$ we have $n = \zero \lor n = 1$. + It suffices to show that for all $n \in \indexset[U]$ we have + for all $m \in \indexset[U]$ such that + $n < m$ we have $\index[U](n) \subseteq \index[U](m)$. + Fix $n \in \indexset[U]$. + Fix $m \in \indexset[U]$. + \begin{byCase} + \caseOf{$n = 1$.} Trivial. + \caseOf{$n = \zero$.} + \begin{byCase} + \caseOf{$m = \zero$.} Trivial. + \caseOf{$m = 1$.} Omitted. + \end{byCase} + \end{byCase} + \end{subproof} + + $A \subseteq X$. + $(X \setminus B) \subseteq X$. + We show that for all $x \in U$ we have $x \subseteq X$. + \begin{subproof} + Omitted. + \end{subproof} + + We show that $\closure{A}{X} \subseteq \interior{(X \setminus B)}{X}$. + \begin{subproof} + Omitted. + \end{subproof} + We show that for all $n,m \in \indexset[U]$ such that $n < m$ we have + $\closure{\index[U](n)}{X} \subseteq \interior{\index[U](m)}{X}$. + \begin{subproof} + Omitted. + \end{subproof} +\end{proof} + + +\begin{proposition}\label{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 + $\closure{A}{X} \subseteq \interior{C}{X} \subseteq \closure{C}{X} \subseteq \interior{B}{X}$. +\end{proposition} +\begin{proof} + Omitted. +\end{proof} + +\begin{proposition}\label{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 $X$. +\end{proposition} +\begin{proof} + + + %\begin{definition}\label{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 $y \subset c \subset x$ + % and for all $g \in C'$ such that $g \neq c$ we have not $y \subset g \subset x$. + %\end{definition} + + + + +\end{proof} + + + + + + + + + % The next thing we need to define is the uniform staircase function. % This function has it's domain in $X$ and maps to the closed interval $[0,1]$. % These functions should behave als follows, @@ -195,7 +312,7 @@ The first tept will be a formalisation of chain constructions. \begin{theorem}\label{urysohn} Let $X$ be a urysohn space. - Suppose $A,B \subseteq \closeds{X}$. + Suppose $A,B \in \closeds{X}$. Suppose $A \inter B$ is empty. There exist $f$ such that $f \in \funs{X}{\intervalclosed{\zero}{1}}$ and $f(A) = 1$ and $f(B)= 0$ and $f$ is continuous. |