On the weak and pointwise topologies in function spaces

2015/04/17 by Mikołaj Krupski, Krupski, Mikołaj
Mathematics · #46E10 #54C35 #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN)

paper · pdf · doi:10.48550/arxiv.1504.04554

openalex publication_date 2015/04/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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