Selections for Paraconvex-valued Mappings on Non-paracompact Domains

We prove that Michael’s paraconvex-valued selection theorem for paracompact spaces remains true for C’(E)-valued mappings defined on collectionwise normal spaces. Some possible generalisations are also given.

💡 Research Summary

The paper revisits Michael’s classical selection theorem for set‑valued mappings whose values are α‑paraconvex subsets of a Banach space, and shows that the paracompactness hypothesis on the domain can be replaced by the much weaker condition of collectionwise normality. The authors work with the family C′α(E)=Cα(E)∪{E}, where Cα(E) consists of all non‑empty closed α‑paraconvex subsets of a Banach space E (0≤α<1). The main result, Theorem 2.1, states that if X is a τ‑collectionwise normal space, E is a Banach space with topological weight w(E)≤τ, and ϕ:X→C′α(E) is lower‑semicontinuous, then (a) ϕ admits a continuous selection, and (b) any continuous approximation g:X→E that stays within a uniform distance r of ϕ can be refined to a genuine selection f with distance bounded by a constant δ·r (δ depends only on α).

The proof proceeds in two stages. First, Proposition 2.2 constructs an increasing closed cover {A_n} of X subordinate to any increasing open cover {V_n} of E, using a u.s.c. multi‑selection ψ:X→C(E) guaranteed by a theorem of Choban and Vălov. This cover allows the authors to localise the problem. Second, Lemma 2.3 (a variant of Michael’s proximal continuity lemma) shows that if ψ is d‑proximal continuous (i.e., both d‑l.s.c. and d‑u.s.c.) and convex‑valued, and ϕ is an l.s.c. multi‑selection of ψ with compact values whenever ϕ≠ψ, then ϕ has a continuous selection.

For part (b) of Theorem 2.1, the authors pick a number γ with α<γ<1 and define a recursive sequence of continuous maps f_n:X→E. At each step they set ψ_{n+1}(x)=B_{γ^n r}(f_n(x)) and ϕ_{n+1}(x)=conv(ϕ(x)∩B_{γ^n r}(f_n(x))). The mapping ϕ_{n+1} is l.s.c., convex‑valued, and a multi‑selection of ψ_{n+1}; Lemma 2.3 then yields a continuous selection f_{n+1} satisfying d(f_n,f_{n+1})≤γ^n r. The sequence {f_n} is a Cauchy (Cauchy‑type) sequence in the complete Banach space, hence converges uniformly to a continuous map f, which is a genuine selection for ϕ and satisfies the required approximation bound.

Part (a) uses the closed cover from Proposition 2.2 to obtain an increasing open cover {U_n} of X. On each U_n the construction from (b) provides a continuous selection f_n that agrees with the previous one on the smaller set, and the union of these local selections yields a global continuous selection for ϕ.

The paper then explores several extensions. Theorem 3.2 and Corollary 3.3 show that the same conclusions hold for τ‑paracompact normal spaces (including countably paracompact normal spaces). Theorem 3.4 and 3.5 replace the constant α by a function h:(0,∞)→