On the weak and pointwise topologies in function spaces

TL;DR

This paper investigates the topological relationship between weak and pointwise topologies on spaces of continuous functions over compact spaces, showing they are not homeomorphic for certain classes of spaces.

Contribution

It proves that for infinite compact metrizable C-spaces, the spaces of continuous functions with weak and pointwise topologies are not homeomorphic, resolving an open question.

Findings

C_w(K) and C_p(K) are not homeomorphic for infinite compact metrizable C-spaces.

The result applies to finite-dimensional compact metrizable spaces.

Addresses an open problem in topology of function spaces.

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 $K$.