Whitney levels in hyperspaces of non-metrizable continua

Abstract

Let X be a Hausdorff continuum (a nondegenerate, compact, connected, Hausdorff space). Let C(X) (respectively F1(X)) denote the hyperspace of its subcontinua (respectively, its one-point sets), endowed with the Vietoris topology. In this paper we introduce the definition of Whitney levels in C(X) and discuss some basic properties. With this definition, the subsets F1(X) and {X} of C(X) are Whitney levels in C(X), so we call them trivial Whitney levels. In the particular case when X is a generalized arc, we give a condition for the existence of non-trivial Whitney levels in its hyperspace of subcontinua. Finally, we apply this result to the study of Whitney levels in C(X) when X is the Long Arc and the Lexicographic Square.