1\over 2-Homogeneous hyperspace suspensions

Abstract

We continue the study of 1 2 -homogeneity of the hyperspace suspension of continua. We prove that if X is a decomposable continuum and its hyperspace suspension is 1 2 -homogeneous, then X must be continuum chainable. We also characterize 1 2 -homogeneity of the hyperspace suspension for several classes of continua, including: continua containing a free arc, atriodic and decomposable continua, and decomposable irreducible continua about a finite set.

Degree of homogeneity on cones

“…An orbit of X is the action of H(X) at a point x 0 of X, namely O X (x 0 ) = {h(x 0 ) : h ∈ H(X)}. Given a positive integer n, a space is said to be 1 n -homogeneous provided that X has exactly n orbits, in which case we say that the degree of homogeneity of X is n. Since 2006 there has been increasing interest in the study of 1 2 -homogeneity, in fact, several papers have been written on the subject: [1,2,6,[10][11][12][13][16][17][18][19][20][21][22][23][24]. Higher degrees of homogeneity appear to be studied only in [8,26].…”