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Diffstat (limited to 'library/topology/urysohn2.tex')
| -rw-r--r-- | library/topology/urysohn2.tex | 30 |
1 files changed, 22 insertions, 8 deletions
diff --git a/library/topology/urysohn2.tex b/library/topology/urysohn2.tex index 40a3615..71de210 100644 --- a/library/topology/urysohn2.tex +++ b/library/topology/urysohn2.tex @@ -68,18 +68,32 @@ Let $H = \carrier[X] \setminus B$. Let $P = \{x \in \pow{X} \mid x = A \lor x = H \lor (x \in \pow{X} \land (\closure{A}{X} \subseteq \interior{U}{X} \subseteq \closure{U}{X} \subseteq \interior{H}{X}))\}$. + Let $\eta = \carrier[X]$. - - - Define $f : X \to \reals$ such that $f(x) = $ + + % Provable + % vvv + Define $F : \eta \to \reals$ such that $F(x) =$ \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 + & \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 \end{cases} - + %Define $f : \naturals \to \pow{P}$ such that $f(x)=$ + %\begin{cases} + % & \emptyset & \text{if} x = \zero \\ + % & \{A, H\} & \text{if} x = 1 \\ + % & 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 $q \mathrel{R} S$. Trivial. |