Metrizable TAP, HTAP and STAP groups

Abstract

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

References in corpus (1)

• NSS and TAP properties in topological groups close to being compact

Cited by in corpus (3)

• Group-valued continuous functions with the topology of pointwise convergence
• NSS and TAP properties in topological groups close to being compact
• Productivity of sequences with respect to a given weight function