We construct two countable spaces having no non-trivial convergent sequences. One space has every point biaccessible (by a countable discrete set) and the other has every point accessible but not biaccessible (by a countable discrete set). We prove that if X is a compact Hausdorff space and $x\in cl_X(A)\setminus A$, then there exists a free sequence $F\subset A$ such that $x \in cl_X(F)$. It follows that in a compact Hausdorff space every non-isolated point is in the closure of a discrete set, and that in a compact Hausdorff space of countable tightness, every non isolated point is biaccessible.