Retracts and retracts by pseudo-deformation of continua without \(R^3\)-sets, \(R^4\)-continua, and s-points (Q6545214)

A continuum is a nondegenerate, compact, connected, metric space. A continuum \(X\) is pseudo-contractible if there exist a mapping \(h\colon X \times T\to X\) and points \(z\in X\) and \(a, b \in T\) such that \(h(x, a) = x\) and \(h(x, b) = z\), for each \(x\in X\). A topological property \(P\) is:\N\begin{itemize}\N\item invariant under retraction if each retract of a continuum having \(P\) has \(P\);\N\item reversible under retraction by pseudo-deformation if the condition a subcontinuum of a continuum \(X\) has \(P\) implies that \(X\) has \(P\).\N\end{itemize}\N\NIt has been proven that pseudo-contractibility is invariant under retractions and is reversible under retractions by pseudo-deformation. At the same time, it is known that each pseudo-contractible continuum has property (b) and that property (b) has a behavior similar to pseudo-contractibility in the following sense. \textit{G. T. Whyburn} [Analytic topology. Providence, RI: American Mathematical Society (AMS) (1942; Zbl 0061.39301)] proved that property (b) is invariant under retractions and \textit{F. Capulín} and \textit{A. C. Sierra-Cuevas} announced in [``Pseudo-deformations on topological spaces'', Preprint (2023)] that the property (b) is reversible under retractions by pseudo-deformation.\N\NOn the other hand, in [\textit{F. Capulín} et al., Glas. Mat., III. Ser. 53, No. 2, 359--370 (2018; Zbl 1423.54035)] and in [\textit{F. Capulín} et al., Topology Appl. 334, Article ID 108552, 13 p. (2023; Zbl 1522.54045)], it was proved that the pseudo-contractibility of a continuum \(X\) implies the absence of \(R^3\)-sets, s-points and \(R^4\)-continua in \(X\). So, the following problem arises.\N\N\textbf{Problem:} To determine if the absence of \(R^3\)-sets, s-points and \(R^4\)-continua in a continuum \(X\) is invariant under retractions and reversible under retractions by pseudo-deformation.\N\NIn the paper under review, the authors solve the problem by proving that the absence of \(R^3\)-sets, the absence of \(R^4\)-continua and the absence of s-points are reversible under retractions by pseudo-deformation, and the absence of \(R^3\)-sets and the absence of \(R^4\)-continua are invariant under retractions while the absence of s-points is not.