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.

# Pontryagin Duality

## See also

Abelian Group, Homeomorphism## Explore with Wolfram|Alpha

## Cite this as:

Weisstein, Eric W. "Pontryagin Duality."
From *MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/PontryaginDuality.html