An infinite game on groups

We consider an infinite game on a group G, defined relative to a subset A of G. The game is denoted G(G,A). The finite version of the game, introduced in [1], was inspired by an attack on the RSA cryptosystem as used in an implementation of SSL. Besides identifying circumstances under which player TWO does not have a winning strategy, we show for the topological group of real numbers that if C is a set of real numbers having a selection property (*) introduced by Gerlits and Nagy, then for any interval J of positive length, TWO has a winning strategy in the game G(ℝ J ∪ C).