## Abstract

We prove that the locally convex space C_{p}(X) of continuous real-valued functions on a Tychonoff space X equipped with the topology of point-wise convergence is distinguished if and only if X is a Δ-space in the sense of Knight in [Trans. Amer. Math. Soc. 339 (1993), pp. 45–60]. As an application of this characterization theorem we obtain the following results: 1) If X is a Čech-complete (in particular, compact) space such that C_{p}(X) is distinguished, then X is scattered. 2) For every separable compact space of the Isbell–Mrówka type X, the space C_{p}(X) is distinguished. 3) If X is the compact space of ordinals [0, ω_{1}], then C_{p}(X) is not distinguished. We observe that the existence of an uncountable separable metrizable space X such that C_{p}(X) is distinguished, is independent of ZFC. We also explore the question to which extent the class of Δ-spaces is invariant under basic topological operations.

Original language | English |
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Pages (from-to) | 86-99 |

Number of pages | 14 |

Journal | Proceedings of the American Mathematical Society, Series B |

Volume | 8 |

DOIs | |

State | Published - 1 Jan 2021 |

## Keywords

- Distinguished locally convex space
- Isbell–Mrówka space
- scattered compact space
- Δ-set

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Geometry and Topology
- Discrete Mathematics and Combinatorics

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