An Infinite - Dimensional Homogeneous Indecomposable Continuum

We prove that every homogeneous continuum is an open retract of a non-metric homogeneous indecomposable continuum. 0. Introduction. A continuum X is indecomposable if it cannot be written as the union of two proper subcontinua. Examples of 1-dimensional homogeneous indecomposable continua are the pseudo-arc and the solenoids. J. T. Rogers asked whether there is an example of a homogeneous indecomposable metric continuum of dimension greater than I (see [1]). The aim of this note is to show that every homogeneous continuum is an open retract of a non-metric homogeneous indecomposable continuum. Consequently, we leave Rogers' question unanswered but prove that the condition on metrizability in his question is essential. 1. The spaces dX. Throughout, X denotes a compact Hausdorff space. In this section we shall associate to X a certain compactum dX that will be the first step in an inverse limit construction later. For every function f • {0,1} X and x • X, let Ix: X -• {0,1} be the function defined by fx(Y) = f(Y), fory•x,