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Preprint

Xabier Domínguez Vaja Tarieladze

2009-09-08

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 \({\rm TAP}\) group is introduced and it is shown in particular that \({\rm NSS}\) groups are \({\rm TAP}\). We prove that conversely, Weil complete metrizable \({\rm TAP}\) groups are \({\rm NSS}\). We define also the narrower class of \({\rm STAP}\) groups, show that the \({\rm NSS}\) groups are in fact \({\rm STAP}\) 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 \({\rm TAP}\) property. We show that for no infinite Tychonoff space \(X\), the group \(C_p(X,\mathbb R)\) has the \({\rm STAP}\) property. We also show that a metrizable locally balanced topological vector group is \({\rm STAP}\) iff it does not contain a subgroup topologically isomorphic to \(\mathbb Z^{(\mathbb N)}\).

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Dmitri Shakhmatov, Jan Spěvák (2009)

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T. M. Hildebrandt (1940)

http://arxiv.org/licenses/nonexclusive-distrib/1.0/