diff options
Diffstat (limited to 'library/topology')
| -rw-r--r-- | library/topology/urysohn.tex | 34 |
1 files changed, 27 insertions, 7 deletions
diff --git a/library/topology/urysohn.tex b/library/topology/urysohn.tex index 028e10f..f34f12f 100644 --- a/library/topology/urysohn.tex +++ b/library/topology/urysohn.tex @@ -5,6 +5,7 @@ \import{function.tex} \import{set.tex} \import{cardinal.tex} +\import{relation.tex} \section{Urysohns Lemma} % In this section we want to proof Urysohns lemma. @@ -188,10 +189,20 @@ The first tept will be a formalisation of chain constructions. \end{lemma} +\begin{axiom}\label{pot1} + $\pot \in \funs{\naturals}{\naturals}$. +\end{axiom} +\begin{axiom}\label{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} + $\pot(n) = \apply{\pot}{n}$. +\end{abbreviation} \begin{proposition}\label{urysohnchain_induction_begin} @@ -356,15 +367,24 @@ The first tept will be a formalisation of chain constructions. and for all $n \in \indexset[\zeta]$ we have $n+1 \in \indexset[\zeta]$ and $\index[\zeta](n+1)$ is a refinmant of $\index[\zeta](n)$. \begin{subproof} - %Let $P = \{ n \in \naturals \mid \exists \zeta. ((n = \zero \land \index[\zeta](n) = \eta \land \index[\zeta](n+1)$ is a refinmant of $\index[\zeta](n)) \lor (n \neq \zero \land \index[\zeta](n)$ is a urysohnchain in $X$ \land $\index[\zeta](n+1)$ is a refinmant of $\index[\zeta](n)))\}$. - %Let $C = \{ x \mid \text{$x$ is a uysohncahin in $X$}\}$. - Let $\alpha = \{x \in C \mid \exists y \in \alpha. x \refine y \lor x = \eta\}$. - Let $\beta = \{ (n,x) \mid n \in \naturals \mid \exists m \in \naturals. \exists y \in \alpha. (x \in \alpha) \land ((x \refine y \land m = n+1 )\lor ((n,x) = (\zero,\eta)))\}$. + %Let $\alpha = \{x \in C \mid \exists y \in \alpha. x \refine y \lor x = \eta\}$. + %Let $\beta = \{ (n,x) \mid n \in \naturals \mid \exists m \in \naturals. \exists y \in \alpha. (x \in \alpha) \land ((x \refine y \land m = n+1 )\lor ((n,x) = (\zero,\eta)))\}$. + % + % % TODO: Proof that \beta is a function which would be used for the indexing. + %For all $n \in \naturals$ we have there exist $x \in \alpha$ such that $(n,x) \in \beta$. + %$\dom{\beta} = \naturals$. + %$\ran{\beta} = \alpha$. + %$\beta \in \funs{\naturals}{\alpha}$. + %Take $\zeta$ such that $\carrier[\zeta] = \alpha$ and $\indexset[\zeta] = \naturals$ and $\index[\zeta] = \beta$. + Omitted. + \end{subproof} - % TODO: Proof that \beta is a function which would be used for the indexing. - + We show that for all $n \in \indexset[\zeta]$ we have $\index[\zeta](n)$ has cardinality $\pot(n)$. + \begin{subproof} + Trivial. \end{subproof} + |