Metrizable Tap and Stap Groups
نویسندگان
چکیده
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), doi:10.1016/j.topol.2009.06.022] the concept of a TAP group is introduced and it is shown in particular that the NSS groups are TAP. We prove that conversely, the Weil complete metrizable TAP groups are NSS. We define also the narrower class of STAP groups, show that the NSS groups are if 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 show also that a metrizable locally balanced topological vector group is STAP iff it does not contain a subgroup topologically isomorphic to Z(N).
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