Topological properties of function spaces Ck(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 if 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 if 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 Frechet Urysohn space if C-k (X, 2) is a Polish space if X is a Polish locally compact space, (5) the space C-k (X, 2) is normal if 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). 2016 Elsevier B.V. All rights reserved.