Inductive Limit Topology and First Countability

Solution 1:

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.

Solution 2:

It seems that usually $X$ is not first countable and even the direct limit $\mathbb{R}^\infty$ of $\mathbb{R^n}$ has the uncountable character, equal to the small cardinal $\mathfrak d$. If you need references, I shall ask my scientific consultant for them.