Metrizable TAP, HTAP and STAP groups

In a recent paper by D. Shakhmatov and J. Sp\v{e}v'ak [Group-valued continuous functions with the topology of pointwise convergence, Topology and its Applications (2009), doi:10.1016/j.topol.2009.06.022] the concept of a $\mathrm{T}\mathrm{A}\mathrm{P}$ group is introduced and it is shown in particular that $\mathrm{N}\mathrm{S}\mathrm{S}$ groups are $\mathrm{T}\mathrm{A}\mathrm{P}$. We prove that conversely, Weil complete metrizable $\mathrm{T}\mathrm{A}\mathrm{P}$ groups are $\mathrm{N}\mathrm{S}\mathrm{S}$. We define also the narrower class of ${\rm STAP}$groups,showthatthe${\rm NSS}$groupsareinfact${\rm STAP}$ 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. Sp\v{e}v'ak asserts: a Tychonoff space $X$ is pseudocompact if and only if $C_p(X,\mathbb{R})$ has the $\mathrm{T}\mathrm{A}\mathrm{P}$ property. We show that for no infinite Tychonoff space $X$, the group $C_p(X,\mathbb{R})$ has the $\mathrm{S}\mathrm{T}\mathrm{A}\mathrm{P}$ property. We also show that a metrizable locally balanced topological vector group is $\mathrm{S}\mathrm{T}\mathrm{A}\mathrm{P}$ iff it does not contain a subgroup topologically isomorphic to ${\mathbb{Z}}^{(\mathbb{N})}$.