Is a topology determined by its convergent sequences?

There is a category of "sequential spaces" in which objects are spaces defined by their convergent sequences and morphisms are those maps which send convergent sequences to convergent sequences.

As stated above, all metric spaces are sequential spaces, but so are all manifolds, all finite topological spaces, and all CW-complexes.

To build this category, one actually just needs to look at the category of right $M$-sets for a certain monoid $M$. Consider first the "convergent sequence space" $S:=${$frac{1}{n}|nin{mathbb N}cup${$infty$}}$subset {mathbb R}$. In other words $S$ is a countable set of points converging to 0, and including $0$. Let $M$ be the monoid of continuous maps $Sto S$ with composition. Then an $M$-set is a "set of convergent sequences" closed under taking subsequences.

The category of $M$-sets is a topos, so it has limits, colimits, function spaces, etc. And every $M$-set has a topological realization which is a sequential space.