Cosmicity of Rω1-factorizable spaces

Determine whether every Rω1-factorizable Tychonoff space is cosmic; specifically, ascertain whether any topological space X for which every continuous function f: X → Rω1 factors through a continuous map into a second-countable space necessarily has a countable network.

Background

The paper introduces Rω1-factorizable spaces: X is Rω1-factorizable if every continuous map f: X → Rω1 factors through a second-countable space. This is equivalent to X × D(ω1) being R-factorizable, and implies several structural properties (hereditarily Lindelöf and hereditarily separable; second-countable if w(X) ≤ ω1). The authors also show that the existence of nonmetrizable Rω1-factorizable spaces is consistent with ZFC under certain set-theoretic assumptions.

Despite these strong constraints, it remains unclear if Rω1-factorizability forces a space to be cosmic (i.e., to have a countable network). Establishing cosmicity would provide a significant structural characterization of Rω1-factorizable spaces beyond Lindelöfness and separability.