2015/04/16 by S. Gabriyelyan, Gabriyelyan, S.
Mathematics · #54C35 #54D50 #54D55 #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #math.FA #math.GN #msc:54C35 #msc:54D50 #msc:54D55
paper · pdf · doi:10.48550/arxiv.1504.04198
arxiv created 2015/04/21 · arxiv updated 2015/04/22
Let X be a zero-dimensional metric space and X' its derived set. We prove the following assertions: (1) the space Ck(X,2) is an Ascoli space iff Ck(X,2) is k_ℝ-space iff either X is locally compact or X is not locally compact but X' is compact, (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' is compact, (3) 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' is compact, (4) Ck(X,2) is a Fréchet--Urysohn space iff Ck(X,2) is a Polish space iff X is a Polish locally compact space, (5) Ck(X,2) is normal iff X' is separable, (6) Ck(X,2) has countable tightness iff X is separable. In cases (1)-(3) we obtain also a topological and algebraical structure of Ck(X,2).