  This is a survey of some of the consequences of the recently introduced
congruences on the theory of connectednesses (radical classes) and
disconnectednesses (semisimple classes) of graphs and topological spaces. In
particular, it is shown that the connectednesses and disconnectednesses can be
obtained as Hoehnke radicals and a connectedness has a characterization in
terms of congruences resembling the classical characterization of its algebraic
counterpart using ideals for a radical class. But this approach has also shown
that there are some unexpected differences and surprises: an ideal-hereditary
Hoehnke radical of topological spaces or graphs need not be a Kurosh-Amitsur
radical and in the category of graphs with no loops, non-trivial
connectednesses and disconnectednesses exist, but all Hoehnke radicals
degenerate.
