Metrizable TAP, HTAP and STAP groups

In a recent paper by D. Shakhmatov and J. Spěvák [D. Shakhmatov, J. Spěvák, Group-valued continuous functions with the topology of pointwise convergence, Topology Appl. 157 (2010) 1518–1540] the concept of a TAP group is introduced and it is shown in particular that NSS groups are TAP. We define the classes of STAP and HTAP groups and show that in general one has the inclusions NSS ⊂ STAP ⊂ HTAP ⊂ TAP. We show that metrizable STAP groups are NSS and that Weil-complete metrizable TAP groups are NSS as well. We prove that an abelian TAP group is HTAP, while, as recently proved by D. Dikranjan and the above mentioned authors, there are nonabelian metrizable TAP groups which are not HTAP. A remarkable characterization of pseudocompact spaces obtained in the above mentioned paper asserts: a Tychonoff space X is pseudocompact if and only if C p ( X , R ) has the TAP property. We show that for no infinite Tychonoff space X, the group C p ( 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 ) .