On the weak and pointwise topologies in function spaces II

TL;DR

This paper investigates the topological relationship between function spaces with weak and pointwise topologies, extending previous results to include finite-dimensional Valdivia compact spaces and addressing an open question about their homeomorphism.

Contribution

It extends Krupski's negative result on the non-homeomorphism of $C_w(K)$ and $C_p(L)$ to include finite-dimensional Valdivia compact spaces, broadening the understanding of their topological distinctions.

Findings

No homeomorphism between $C_w(K)$ and $C_p(L)$ when $K$ and $L$ are infinite compact spaces with the specified properties.

Extension of Krupski's result to finite-dimensional Valdivia compact spaces.

Clarification of the topological differences between weak and pointwise function spaces in this class.

Abstract

For a compact space $K$ we denote by $C_w(K)$ ($C_p(K)$) the space of continuous real-valued functions on $K$ endowed with the weak (pointwise) topology. In this paper we discuss the following basic question which seems to be open: Let $K$ and $L$ be infinite compact spaces. Can it happen that $C_w(K)$ and $C_p(L)$ are homeomorphic? M. Krupski proved that the above problem has a negative answer when $K=L$ and $K$ is finite-dimensional and metrizable. We extend this result to the class of finite-dimensional Valdivia compact spaces $K$.