diff options
| -rw-r--r-- | library/topology/separation.tex | 14 |
1 files changed, 9 insertions, 5 deletions
diff --git a/library/topology/separation.tex b/library/topology/separation.tex index f055495..530a51f 100644 --- a/library/topology/separation.tex +++ b/library/topology/separation.tex @@ -99,10 +99,6 @@ For all $y \in \unions{V}$ we have $y \in \carrier[X] \setminus \{x\}$. Follows by set extensionality. \end{subproof} - - - - % Let $U_{y} \in \opens[X]$ such that $x \neq y \in X$. Error: unexpected '{' expecting digit %For all $y$ there exists $U$ such that $x \neq y$ and $y \in U$ and $U \in \opens[X]$. @@ -149,5 +145,13 @@ Then $X$ is a \teeone-space. \end{proposition} \begin{proof} - Omitted. % TODO + We show that for all $x,y\in\carrier[X]$ such that $x\neq y$ + there exist $U, V\in\opens[X]$ such that + $U\ni x\notin V$ and $V\ni y\notin U$. + \begin{subproof} + $X$ is hausdorff. + For all $x,y\in\carrier[X]$ such that $x\neq y$ + there exist $U, V\in\opens[X]$ such that + $x\in U$ and $y\in V$ and $U$ is disjoint from $V$. + \end{subproof} \end{proof} |