Aha, so a space which has a countable basis and which can simultaneously separate any countable collection of isolated points must be regular. You look at any closed C and point x which by hypothesis can't be separated, choose a chain of neighborhoods U_i of x such that one of them fits inside any neighborhood of x, realize that they must meet successive V_i each owning a c_i of C, and then observe by sigma-s.s. that we must be able to find a c_j in the closure of infintely many other c_i (i ne j) so it has to be in the closure of infinitely (and cofinally!) many U_i, violating Hausdorff, QED.