While preparing a course in complex analysis, I stumbled over a remark in Dudziak's book on removable sets, namely that any totally disconnected $K \subset\subset {\mathbb C}$ must have a connected complement; a remark, "that verifying the reader may find one of those exercises in 'mere' point-set topology that is a wee bit frustrating". Out of curiosity I spent an evening with this question. The assertion turned out to be a simple consequence of Theorem 14.2 ("If $x$ and $y$ are separated by the closed set $F$ in the open or closed plane they are separated by a component of $F$.") in an old book "Elements of the Topology of Plane Sets of Points" by M.H.A. Newman, Cambridge 1951. There, the proof is based on an lemma by Alexander (used in his proof of the Jordan-Brouwer separation theorem; Trans. AMS 23, 333-349, 1922), stating that, for disjoint closed sets $F_1$ and $F_2$ in the plane, two points which are connected in the complement of $F_1$ and in the complement of $F_2$ are connected in the complement of $F_1 \cup F_2$. This lemma fails for more general surfaces (Newman gives a counterexample for the torus) and is proved by homological methods. So, here are my questions:

(a) is there a more modern reference to these kind of results (Newman's book uses quite an ideosyncratic terminology and notation);

(b) is there a simpler proof not using Alexander's lemma (or similarily deep results);

(c) how about the connectedness of the complement of a totally disconnected closed set in surfaces (or topological spaces) more general than the plane?