## Abstract

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).

Original language | English |
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Pages (from-to) | 335-346 |

Number of pages | 12 |

Journal | Topology and its Applications |

Volume | 209 |

DOIs | |

State | Published - 15 Aug 2016 |

## Keywords

- Ascoli space
- Fréchet–Urysohn
- Function space
- Metric space
- Sequential
- k-Space
- k-Space

## All Science Journal Classification (ASJC) codes

- Geometry and Topology

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