可计量的TAP、HTAP和STAP组
Metrizable TAP, HTAP and STAP groups
摘要：
在D. Shakhmatov和J. Spanovák [Group-valued continuous functions with the点式收敛拓扑，Topology and its Applications（2009），doi：10.1016/j.topol.2009.06.022]引入了${\rm TAP}$群的概念，并且特别地示出了${\rm NSS}$群是${\rm TAP}$。反过来，我们证明了Weil完全可度量化的${\rm TAP}$群是${\rm NSS}$ .我们还定义了较窄的一类${\rm STAP}$群，证明了${\rm NSS}$群实际上是${\rm STAP}$，并且在可度量化的情况下，逆命题成立. D. Shakhmatov和J. Spanvák断言：一个吉洪诺夫空间$X$是伪紧的当且仅当$C_p(X,\mathbb R)$具有${\rm TAP}$性质。我们证明了对于没有无限的吉洪诺夫空间$X$，群$C_p(X,\mathbb R)$具有${\rm STAP}$性质.我们还证明了可度量化的局部平衡拓扑向量群是${\rm STAP}$当且仅当它不包含拓扑同构于$\mathbb Z^{(\mathbb N)}$的子群。
英文摘要：
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 $C_p(X,\mathbb R)$ 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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