Topological properties of function spaces Ck(X,2) over zero-dimensional metric spaces X

Let X   be a zero-dimensional metric space and X′X′ its derived set. We prove the following assertions: (1) the space Ck(X,2)Ck(X,2) is an Ascoli space iff Ck(X,2)Ck(X,2) is kRkR-space iff either X is locally compact or X   is not locally compact but X′X′ is compact, (2) Ck(X,2)Ck(X,2) is a k-space iff either X is a topological sum of a Polish locally compact space and a discrete space or X   is not locally compact but X′X′ is compact, (3) Ck(X,2)Ck(X,2) is a sequential space iff X is a Polish space and either X is locally compact or X   is not locally compact but X′X′ is compact, (4) Ck(X,2)Ck(X,2) is a Fréchet–Urysohn space iff Ck(X,2)Ck(X,2) is a Polish space iff X   is a Polish locally compact space, (5) the space Ck(X,2)Ck(X,2) is normal iff X′X′ is separable, (6) Ck(X,2)Ck(X,2) has countable tightness iff X   is separable. In cases (1)–(3) we obtain also a topological and algebraic structure of Ck(X,2)Ck(X,2).