Topological connectednesses and congruences

Abstract

It is known that the connectednesses of topological spaces in the sense of Preuβ is the topological analogue of the Kurosh-Amistsur radicals of algebraic structures in a categorical sense. Here this connection is further explored. As in universal algebra, a congruence on a topological space has been defined. It is shown that a connectedness can be characterized in terms of conditions on congruences which are the precise topological analogues of those conditions that characterize the radical classes of rings in terms of ideals.