gn.general topology - Sequential topological vector spaces

The space of all functions $mathbb Rtomathbb R$ with the topology of pointwise convergence is obviously sequential but not first countable.

Indeed, for every $xinmathbb R$, the set $U_x:={f:|f(x)|<1}$ is open. If the space had a countable base of neighborhoods of 0, every set of the form $U_x$ would contain an element of the base. So some element $U$ of the base would be contained in infinitely many sets $U_{x_1}, U_{x_2}, dots$ and hence in the intersection $V=bigcap U_{x_i}$. But 0 is not in the interior of $V$ because there is a sequence of functions outside $V$ that pointwise converges to 0. Namely the $n$th member of the sequence is the characteristic function of the set ${x_i:ige n}$.