 Topology with countable closed sets away from one point

Fix a point $a$ of an uncountable set $X$ and declare $F$ closed when $a\in F$ or $F$ is countable. Finite unions and arbitrary intersections satisfy the closed-set axioms. Every convergent sequence is eventually constant: the countable set of its values different from the limit is closed, so its complement is a neighborhood of the limit. A real-valued map constant at $f(a)$ outside a countable set is continuous, because each closed inverse image either contains $a$ or lies in that countable exceptional set.