diff options
Diffstat (limited to 'library')
| -rw-r--r-- | library/topology/urysohn2.tex | 14 |
1 files changed, 12 insertions, 2 deletions
diff --git a/library/topology/urysohn2.tex b/library/topology/urysohn2.tex index a64fa7e..40a3615 100644 --- a/library/topology/urysohn2.tex +++ b/library/topology/urysohn2.tex @@ -46,7 +46,13 @@ $\intervalclosed{a}{b} = \{x \in \reals \mid a \leq x \leq b\}$. \end{definition} - +\begin{theorem}\label{urysohnsetinbeetween} + Let $X$ be a urysohn space. + Suppose $A,B \in \closeds{X}$. + Suppose $\closure{A}{X} \subseteq \interior{B}{X}$. + Suppose $\carrier[X]$ is inhabited. + There exist $U \subseteq \carrier[X]$ such that $U$ is closed in $X$ and $\closure{A}{X} \subseteq \interior{U}{X} \subseteq \closure{U}{X} \subseteq \interior{B}{X}$. +\end{theorem} @@ -60,6 +66,10 @@ \end{theorem} \begin{proof} + Let $H = \carrier[X] \setminus B$. + Let $P = \{x \in \pow{X} \mid x = A \lor x = H \lor (x \in \pow{X} \land (\closure{A}{X} \subseteq \interior{U}{X} \subseteq \closure{U}{X} \subseteq \interior{H}{X}))\}$. + + Define $f : X \to \reals$ such that $f(x) = $ \begin{cases} @@ -70,7 +80,7 @@ \end{cases} - $U_1$ + Trivial. |