On the Pytkeev property in spaces of continuous functions (II)

TL;DR

This paper investigates the Pytkeev property in spaces of continuous functions, showing that for Polish spaces, these function spaces with the compact-open topology nearly attain metrizability, and also explores the property under pointwise convergence.

Contribution

It establishes that C(X) for Polish spaces has a strong Pytkeev property with the compact-open topology, advancing understanding of its topological structure.

Findings

C(X) with compact-open topology satisfies a strong Pytkeev property for Polish X

C(X) is nearly metrizable despite not necessarily being metrizable

The Pytkeev property is also analyzed for the pointwise convergence topology

Abstract

We prove that for each Polish space X, the space C(X) of continuous real-valued functions on X satisfies a strong version of the Pytkeev property, if endowed with the compact-open topology. (This shows that whereas it need not be metrizable, it is "very close" to that.) We also consider the Pytkeev property in the case where C(X) is endowed with the topology of pointwise convergence.