On the weak and pointwise topologies in function spaces

Mikołaj Krupski

paper

On the weak and pointwise topologies in function spaces

arXiv:1504.04554

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 address the following basic question which seems to be open: Suppose that $K$ is an infinite (metrizable) compact space. Is it true that $C_{w} (K)$ and $C_{p} (K)$ are homeomorphic? We show that the answer is "no", provided $K$ is an infinite compact metrizable $C$ -space. In particular our proof works for any infinite compact metrizable finite-dimemsional space $K$ .