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 $\ast :M\times M\to M$ such that for every $x,y\in M$ the inclusion $x\ast y\in X$ holds if and only if $x,y\in X$. Assume also that there exist continuous unary operations $u,v:M\to M$ such that $x=u(x)\ast v(x)$ for all $x\in M$. Given a $2^{\omega }$-stable ${\mathbf{\Pi }}_2^0$-hereditary weakly ${\mathbf{\Sigma }}_2^0$-additive class of spaces $\mathcal{C}$, we prove that the pair $(M,X)$ is strongly $({\mathbf{\Pi }}_1^0\cap \mathcal{C},\mathcal{C})$-universal if and only if for any compact space $K\in \mathcal{C}$, subspace $C\in \mathcal{C}$ of $K$ and nonempty open set $U\subseteq M$ there exists a continuous map $f:K\to U$ such that $f^{-1}[X]=C$. This characterization is applied to detecting strongly universal Lawson semilattices.

13 pages

References in corpus (3)

• Wilder continua and their subfamilies as coanalytic absorbers
• More absorbers in hyperspaces
• Hyperspaces of continua with connected boundaries in $\pi$-Euclidean Peano continua