Two classes of spaces reflexive in the sense of Pontryagin

The Pontryagin–van Kampen duality for locally compact Abelian groups can be generalized in two ways to wider classes of topological Abelian groups: in the first approach the dual group $X^\bullet$ is endowed with the topology of uniform convergence on compact subsets of $X$ and in the second, with the topology of uniform convergence on totally bounded subsets of $X$. The corresponding two classes of groups “reflexive in the sense of Pontryagin–van Kampen” are very wide and are so close to each other that it was unclear until recently whether they coincide or not. A series of counterexamples constructed in this paper shows that these classes do not coincide and also answer several other questions arising in this theory. The results of the paper can be interpreted as evidence that the second approach to the generalization of the Pontryagin duality is more natural.