Pontryagin Duality

Let  be a locally compact Abelian group. Let  be the group of all continuous homeomorphisms , in the compact open topology. Then  is also a locally compact Abelian group, where the asterisk defines a contravariant equivalence of the category of locally compact Abelian groups with itself. The natural mapping , sending  to , where , is an isomorphism and a homeomorphism. Under this equivalence, compact groups are sent to discrete groups and vice versa.