The uniform convergence topology on separable subsets

TL;DR

This paper investigates the topological properties of the space of continuous functions on a space X under the uniform convergence topology on separable subsets, including density, closedness, Baire property, and cellularity.

Contribution

It characterizes when the subspace of continuous functions is dense or closed, and analyzes the Baire property and cellularity in these function spaces.

Findings

Cs(X) is dense or closed under certain conditions.

Cs([0,α]) has specific cellularity properties.

Results on the Baire property in Cs(X).

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,{\alpha}]) where [0,{\alpha}] is the space of ordinal numbers belonging to {\alpha} + 1 with its usual order topology.