TITLE: Inductive Limit Topology and First Countability
QUESTION [11 upvotes]: Motivation: I am doing functional analysis on locally convex spaces for the first time and I would like to know when I am allowed to characterise limit points and continuity sequentially. (This may well be a silly question.)
I phrase my question abstractly, but I mostly care about $\mathcal{C}_0^\infty(\mathbb{R}^n)$.
Let $(X_n)$ be an increasing sequence of first-countable locally convex topological vector spaces and let $X=\cup_n X_n$.
If we topologise $X$ with the finest locally convex topology such that the inclusions $X_n\rightarrow X$ are all continuous, is $X$ necessarily a first-countable space?
(A local base for such a topology is given by the the collection of all balanced, convex, absorbent sets whose intersections with every $X_j$ is open in $X_j$.)
I am currently reading from Reed & Simon Methods of Mathematical Physics, I cannot seem to find too many modern and systematic treatments of locally convex vector spaces!

REPLY [13 votes]: One can sharpen Alex Ravsky's answer:
Recall that a topological vector space is metrizable if and only if it is first countable.

Let $X_n \subsetneqq X_{n+1} \subsetneqq \cdots$ be a strictly increasing sequence of Fréchet spaces such that each $X_n$ carries the topology induced by $X_{n+1}$. Then $X = \varinjlim X_n$ is not metrizable.

The point is that $X$ is complete (see e.g. Schaefer-Wolf, Proposition 6.6 and its corollary). If $X$ were metrizable, then $X = \bigcup_{n=1}^\infty X_n$ would be a union of countably many nowhere dense sets, contradicting the Baire category theorem.