If I have a decreasing sequence (infinite, at least countable) of subspaces of topological vector spaces:
$$\ldots E_{n+1} \subseteq E_n \subseteq \ldots \subseteq E_2 \subseteq E_1$$
What is the topology of the intersection?
$$ E = \cap_{i}E_i $$
considering that the topology of $E_n$ is the subspace topology induced by $E_{n-1}$. What I'm thinking is that a possible subbase of the topology is $\{U_i\cap E: U_i \mbox{ is open on }E_i\}$. Any thoughts?
The subspace topology on $E_n$ induced by $E_{n-1}$ is the same as the subspace topology on $E_n$ induced by $E_1$. So the most reasonable choice for $E$ would be the subspace topology induced by $E_1$.