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We present a series of examples of precompact, noncompact, reflexive topological Abelian groups. Some of them are pseudocompact or even countably compact, but we show that there exist precompact non-pseudocompact reflexive groups as well. It is also proved that every pseudocompact Abelian group is a quotient of a reflexive pseudocompact group with respect to a closed reflexive pseudocompact subgroup.

Abstract. We introduce a notion of a Schwartz group, which turns out to be coherent with the well known concept of a Schwartz topological vector space. We establish several basic properties of Schwartz groups and show that free topological Abelian groups, as well as free locally convex spaces, over hemicompact k-spaces are Schwartz groups. We also prove that every hemicompact k-space topological group, in particular the Pontryagin dual of a metrizable topological group, is a Schwartz group.

In a recent paper by D. Shakhmatov and J. Spěvák [Group-valued continuous functions with the topology of pointwise convergence, Topology and its Applications (2009), ] the concept of a TAP group is introduced and it is shown in particular that NSS groups are TAP. We prove that conversely, Weil complete metrizable TAP groups are NSS. We define also the narrower class of STAP groups, show that the NSS groups are in fact STAP and that the converse statement is true in metrizable case. A remarkable characterization of pseudocompact spaces obtained in the paper by D. Shakhmatov and J. Spěvák asserts: a Tychonoff space X is pseudocompact if and only if Cp(X, R) has the TAP property. We show that for no infinite Tychonoff space X, the group Cp(X, R) has the STAP property. We also show that a metrizable locally balanced topological vector group is STAP iff it does not contain a subgroup topologically isomorphic to Z (N) . 0 2000 Mathematics Subject Classification: 22A05, 46A11.