Try the wikipedia entry on sequential spaces. I quote:
"Many conditions have been shown to be equivalent to $X$ being sequential. Here are a few:
$X$ is the quotient of a first countable space.
$X$ is the quotient of a metric space.
For every topological space $Y$ and every map $f : X \to Y$, we have that f is continuous if and only if for every sequence of points (xn) in X converging to x, we have (f(xn)) converging to f(x).
The final equivalent condition shows that the class of sequential spaces consists precisely of those spaces whose topological structure is determined by convergent sequences in the space."
The point is that given a set of test maps from $I$ to a set $X$ you can put a topology on $X$ as the quotient of the disjoint union of copies of $I$ one for each test map. And then $X$ will be the quotient of a first countable space.