Assume $E_1\subset E_2\subset ...$ is a sequence of Frechet spaces, each one embedded in the next. Let $E$ be their union, and one puts the LF topology on $E$.
I wished one can say the following: If $U\subset E$ satisfies that for every $k$, $U\cap E_k$ is open in $E_k$, then $U$ is open in $E$.
But from the definition, one needs to show that for each $x\in U$, there is a balanced, convex set $W$ such that $x+W\subset U$, and $W\cap E_k$ is open for every $k$. Just from the information that $U\cap E_k$ is open it seems difficult to find such $W$. So is the above statement even true?