A rigid space $X$ for which $X\times X$ is homogeneous;\ an application of infinite-dimensional topology

Abstract

Abstract. We give an example of a rigid (= no autohomeomorphisms beyond the identity) space X such that X X X is homogeneous. In fact, X X X is homeomorphic to the Hilbert cube. This answers a question of A. V. Arhangel'skil.1. Introduction. In his survey paper [1, p. 59], A. V. Arhangel'skil asks whether there is an example of a nonhomogeneous (compact) space whose square is homogeneous. The aim of this note is to show that recent results in infinite-dimensional topology can be used to construct a rigid (= no autohomeomorphisms beyond the identity) space X such that X X X sa Q, the Hilbert cube. As a consequence, I X lis homogeneous. Modulo some very nontrivial known facts it is not difficult to construct X. Since X will prevent general topologists from asking certain natural questions concerning homogeneity, we feel that our example should

Uncountable Family of 0-Rigid Continua that are Homeomorphic to Their Inverse Limits

“…The rigidity of topological spaces has been studied in various ways by different authors, e.g., [1][2][3][4][5][6][7][8][10][11][12]. Cook continua are basic examples of non-degenerate continua that are rigid, and therefore, they play an important role in the study of rigid continua, also in continuum theory and dynamical systems in general.…”