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

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