The hyperspace of T-closed subcontinua

Abstract

The concept of T -closed set of a continuum was defined and studied by D.P. Bellamy et al., where T denotes the Jones’s set function. Given a continuum X, in this paper, we define the hyperspace CT(X) as the collection of all T -closed subcontinua of X. We present examples of continua X for which this hyperspace has n elements for each positive integer n or it is the closure of the harmonic sequence or it is the Cantor set. We study the connectedness, compactness, and density of C. Finally, we relate the continuity of T with the structure of this hyperspace.