The uniform convergence topology on separable subsets

2023/07/04 by Jorge Antonio Cruz Chapital, Alejandro Darío Rojas Sánchez, Chapital, Jorge Antonio Cruz +5
Mathematics · #54A25 #54C35 #54E52 #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN)

paper · pdf · doi:10.48550/arxiv.2307.01823

openalex publication_date 2023/07/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a topological space X, let (RX)s := (RX,Ts) be the cartesian product of |X| copies of the real line R with the topology of the uniform convergence on separable subsets of X. In this article we analyze the subspace C(X) of (RX)s of all real-valued continuous functions on X, denoted by Cs(X). We determine when Cs(X) is dense and when is closed in (RX)s, and we obtain some results about the Baire property in Cs(X). Finally, we determine the cellularity of Cs([0,α]) where [0,α] is the space of ordinal numbers belonging to α + 1 with its usual order topology.

Related

• Uniformity of uniform convergence on the family of sets
• Compactness in function spaces
• Classification of spaces of Continuous Function on ordinals
• Baire property of spaces of [0,1]-valued continuous functions
• Spaces not distinguishing ideal pointwise and σ-uniform convergence