It is well known that $\mathcal{D}(\Omega)$ and $\mathcal{D}'(\Omega)$ are not metrizable, and that a topological vector space is metrizable if and only if it is first-countable (Rudin, Thm. 1.24). This doesn't necessarily mean that sequences will not specify the topology of the space, but it isn't good news at all on that front. Nonetheless, very often I've seen "sequence arguments," which attempt (always successfully) to explore the topology of the space using sequences (for example, in "Introduction to the Theory of Distributions" by Friedlander and Joshi, sequences are used to show that, when considered as distributions, the space of smooth, compactly supported functions is dense in $\mathcal{D}'(\Omega)$). I am wondering if some such arguments are inevitably doomed to fail because sequences don't always fully specify the topology.
I think there are two entwined questions here:
Given any set $A$ in one of these spaces, and any limit point $a$ of it, is there a sequence $\{a_n\}$ of elements of $A$ which converges to $a$?
Given any set $A$ in one of these spaces such that all sequences converging to points in $A$ are eventually contained in $A$, must $A$ be open?
I am not sure if affirmative answers to either one of these questions implies an affirmative answer to the other.
A follow-up question (if the answer is affirmative) I have is that if a topological vector space $E$ is the strict locally convex inductive limit of a countable collection of Frechet spaces $E_{n}$ such that $E_n$ is always a closed subspace of $E_{n+1}$, then can the topology of the space be fully specified by sequences?
Any help would be greatly appreciated!
One does NOT understand $\mathscr D$ and $\mathscr D'$ by considering only sequences! For example (I think this was already noted in Hörmander's On the range of convolution operators (Ann. of Math. (2) 76, 1962, 148–170) it is a big difference for an injective operator $\mathscr D \to \mathscr D$ if it has a continuous invers (from its range back to $\mathscr D$) or only a sequentially continuous inverse.
As far as I remember V.B. Moscatelli worked on (perhaps even characterized) inductive limits of Frechet spaces such that every sequentiall closed subspace is closed. A nice article about the interplay of such general questions and analytical problems is Floret, K. (1980). Some aspects of the theory of locally convex inductive limits, Functional Analysis: Surveys and Recent Results II, (K.D. Bierstedt, B. Fuchssteiner, eds.), North Holland Math. Studies 38, Amsterdam, 205-237.