2009/09/08 by Xabier Domínguez Vaja Tarieladze, Tarieladze, Xabier Domínguez Vaja · 1 citation
Computer Science · Mathematics · Medicine · #22A05 #46A11 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR) #Pituitary Gland Disorders and Treatments #math.GN #math.GR #msc:22A05 #msc:46A11
paper · pdf · doi:10.48550/arxiv.0909.1400
openalex publication_date 2009/09/08 · arxiv created 2009/12/01 · arxiv updated 2009/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \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. Spěvák asserts: a Tychonoff space X is pseudocompact if and only if Cp(X,\mathbb R) has the \rm TAP property. We show that for no infinite Tychonoff space X, the group Cp(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).