Metrizable TAP, HTAP and STAP groups - Mathematics > General TopologyReport as inadecuate

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Abstract: In a recent paper by D. Shakhmatov and J. Sp\v{e}v\-ak Group-valuedcontinuous functions with the topology of pointwise convergence, Topology andits Applications 2009, doi:10.1016-j.topol.2009.06.022 the concept of a${ m TAP}$ group is introduced and it is shown in particular that ${ m NSS}$groups are ${ m TAP}$. We prove that conversely, Weil complete metrizable${ m TAP}$ groups are ${ m NSS}$. We define also the narrower class of ${ mSTAP}$ groups, show that the ${ m NSS}$ groups are in fact ${ m STAP}$ andthat the converse statement is true in metrizable case. A remarkablecharacterization of pseudocompact spaces obtained in the paper by D. Shakhmatovand J. Sp\v{e}v\-ak asserts: a Tychonoff space $X$ is pseudocompact if and onlyif $C pX,\mathbb R$ has the ${ m TAP}$ property. We show that for noinfinite Tychonoff space $X$, the group $C pX,\mathbb R$ has the ${ m STAP}$property. We also show that a metrizable locally balanced topological vectorgroup is ${ m STAP}$ iff it does not contain a subgroup topologicallyisomorphic to $\mathbb Z^{\mathbb N}$.

Author: Xabier Domínguez Vaja Tarieladze



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