The strong universality of ANRs with a suitable algebraic structure

Abstract

Let M be an ANR space and X be a homotopy dense subspace in M . Assume that M admits a continuous binary operation * : M × M → M such that for every x, y ∈ M the inclusion x * y ∈ X holds if and only if x, y ∈ X. Assume also that there exist continuous unary operations u, v : M → M such that x = u(x) * v(x) for all x ∈ M . Given a 2 ω -stable Π 0 2 -hereditary weakly Σ 0 2 -additive class of spaces C, we prove that the pair (M, X) is strongly (Π 0 1 ∩ C, C)-universal if and only if for any compact space K ∈ C, subspace C ∈ C of K and nonempty open set U ⊆ M there exists a continuous map f :This characterization is applied to detecting strongly universal Lawson semilattices.