This paper is an exposition of the theory of the hyperspaces 2^X and C(X) of a topological space X. These spaces are obtained from X by collecting the nonempty closed and nonempty closed connected subsets respectively, and are topologized by the Vietoris topology. The paper is organized in terms of increasing specialization of spaces, beginning with T1 spaces and proceeding through compact spaces, compact metric spaces and metric continua. Several basic techniques in hyperspace theory are discussed, and these techniques are applied to elucidate the topological structure of hyperspaces.

The present thesis is devoted to the construction of a probability measure which counts the pairs of closed commuting subgroups in infinite groups. This measure turns out to be an extension of what was known in the finite case as subgroup commutativity degree and opens a new approach of study for the class of near abelian groups, recently introduced in [24, 27]. The extremal case of probability one characterises the topologically quasihamiltonian groups, studied originally by K. Iwasawa [30, 31] in the abstract case and then by F. K¨ummich [35, 36, 37], C. Scheiderer [45, 46], P. Diaconis [11] and S. Strunkov [48] in the topological case. Our probability measure turns out to be a useful tool in describing the distance of a profinite group from being topologically quasihamiltonian. We have been inspired by an idea of H. Heyer in the present context of investigation and in fact we generalise some of his techniques, in order to construct a probability measure on the space of closed subgroups of a profinite group. This has been possible because the space of closed subgroups of a profinite group may be approximated by finite spaces and the consequence is that our probability measure may be approximated by finite probability measures. While we have a satisfactory description for profinite groups and compact groups, the case of locally compact groups remains open in its generality.