Peter Freyd writes on Pontrjagin duality. I would like to mention the generalisation given in (with P.J. HIGGINS and S.A. MORRIS), ``Countable products of lines and circles: their closed subgroups, quotients and duality properties'', {\em Math. Proc. Camb. Phil. Soc.} 78 (1975) 19-32. One point made is that a duality is not necessarily inherited by closed subgroups and Hausdorff quotients. If it is, it is called a strong duality. Then strong duality is inherited by closed subgroups and Hausdorff quotients! I have taught the classification of closed subgroups of R^n in an analysis course. It is a nice result, and the sums you can set use duality in a nice way - treatment borrowed from Bourbaki. Ronnie Brown
Next is Pontryagin's: the category of locally compact groups. The original Pontryagin duality easily generalizes: the category of locally compact modules over a given commutative ring is self-dual. (In the non-commutative case one also obtains a duality but not a self-duality -- unless, of course, the ring is self-dual.) A corollary is that the category of discrete left R-modules is dual to the category of compact right R-modules