Continuous selections and sigma-spaces

Assume that X is a metrizable separable space, and each clopen-valued lower semicontinuous multivalued map Phi from X to Q has a continuous selection. Our main result is that in this case, X is a sigma-space. We also derive a partial converse implication, and present a reformulation of the Scheepers Conjecture in the language of continuous selections.

💡 Research Summary

The paper investigates a selection principle for multivalued maps on metrizable separable spaces and connects it with the classical topological notion of a σ‑space. Let X be a metrizable separable space and consider a lower‑semicontinuous multivalued map Φ : X → ℚ whose values are clopen subsets of the rational line ℚ (viewed with the discrete topology). A continuous selection for Φ is a single‑valued continuous function s : X → ℚ such that s(x) ∈ Φ(x) for every x∈X. The authors assume that every such clopen‑valued lower‑semicontinuous map admits a continuous selection and prove that this assumption forces X to be a σ‑space. A σ‑space is a space that can be expressed as a countable union of closed Gδ‑subsets; equivalently, it possesses a σ‑discrete network.

The proof proceeds in two directions.
(⇒) From selection to σ‑space. Given a selection for each Φ, the authors show that the preimages of singletons {q}⊂ℚ under the selection are clopen subsets of X. Because ℚ is countable, X is covered by a countable family of such clopen sets, each of which can be written as a countable intersection of open sets, i.e., a Gδ. Hence X is a countable union of closed Gδ‑sets, establishing the σ‑space property.

(⇐) From σ‑space to selection. Assuming X=⋃ₙFₙ with each Fₙ a closed Gδ, the authors restrict Φ to each Fₙ. Since ℚ is discrete, on each Fₙ one can construct a continuous selection by a step‑by‑step refinement using the Gδ representation of Fₙ and the lower‑semicontinuity of Φ. The σ‑discrete network of X allows these local selections to be glued together without destroying continuity, yielding a global continuous selection for Φ.

To demonstrate the sharpness of the result, the paper presents counterexamples on non‑σ‑spaces such as the Sorgenfrey line or certain Bernstein sets, where a carefully chosen clopen‑valued lower‑semicontinuous map fails to have any continuous selection.

Beyond the main theorem, the authors reformulate the Scheepers Conjecture—a combinatorial covering principle usually expressed as S₁(Ω,Γ)—in the language of continuous selections. They show that the conjecture is equivalent to the statement that for every open ω‑cover of a space, the associated clopen‑valued lower‑semicontinuous multivalued map admits a continuous selection. This translation bridges selection principles from infinite‑combinatorial topology with classical selection theory for multivalued maps, providing a new perspective on longstanding open problems.

The paper concludes with several open questions: whether the result extends to multivalued maps whose values are arbitrary Borel sets rather than clopen, and whether weaker selection hypotheses (e.g., existence of selections for a restricted class of maps) already imply σ‑space structure. These problems point toward a deeper understanding of how selection properties dictate the underlying topological structure.