Topological properties of function spaces over ordinals

2016/06/13 by Saak Gabriyelyan, Gabriyelyan, Saak, Jan Grebik +5
Mathematics · #54E18 #54F05 #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #Primary 54C35 #Secondary 46A08 #math.FA #math.GN #msc:46A08 #msc:54C35 #msc:54E18 #msc:54F05

paper · pdf · doi:10.48550/arxiv.1606.04025

5 pages, accepted for publication in Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas

arxiv created 2016/11/17 · arxiv updated 2016/11/18

Abstract

A topological space X is said to be an Ascoli space if any compact subset K of C_k(X) is evenly continuous. This definition is motivated by the classical Ascoli theorem. We study the k_R-property and the Ascoli property of C_p(κ) and C_k(κ) over ordinals κ. We prove that C_p(κ) is always an Ascoli space, while C_p(κ) is a k_R-space iff the cofinality of κ is countable. In particular, this provides the first C_p-example of an Ascoli space which is not a k_R-space, namely C_p(ω₁). We show that C_k(κ) is Ascoli iff cf(κ) is countable iff C_k(κ) is metrizable.

Related

• The Ascoli property for function spaces
• The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces
• Ascoli and sequentially Ascoli spaces
• Ascoli's theorem for pseudocompact spaces
• Topological properties of function spaces C_k(X,2) over zero-dimensional metric spaces X