Metrizable TAP, HTAP and STAP groups

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Abstract

In a recent paper by D. Shakhmatov and J. Sp\v{e}v\'ak [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. Sp\v{e}v\'ak 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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Group-valued continuous functions with the topology of pointwise convergence

Dmitri Shakhmatov, Jan Spěvák (2009)

We denote by C_p(X,G) the group of all continuous functions from a space X to a topological group G endowed with the topology of pointwise convergence. We say that spaces X and Y are G-equivalent provided that the topological groups C_p(X,G) and C_p(Y,G) are topologically isomorphic. We investigate which topological properties are preserved by G-equivalence, with a special emphasis being placed on characterizing topological properties of X in terms of those of C_p(X,G). Since R-equivalence coincides with l-equivalence, this line of research "includes" major topics of the classical C_p-theory of Arhangel'skii as a particular case (when G = R). We introduce a new class of TAP groups that contains all groups having no small subgroups (NSS groups). We prove that: (i) for a given NSS group G, a G-regular space X is pseudocompact if and only if C_p(X,G) is TAP, and (ii) for a metrizable NSS group G, a G^*-regular space X is compact if and only if C_p(X,G) is a TAP group of countable tightness. In particular, a Tychonoff space X is pseudocompact (compact) if and only if C_p(X,R) is a TAP group (of countable tightness). We show that Tychonoff spaces X and Y are T-equivalent if and only if their free precompact Abelian groups are topologically isomorphic, where T stays for the quotient group R/Z. As a corollary, we obtain that T-equivalence implies G-equivalence for every Abelian precompact group G. We establish that T-equivalence preserves the following topological properties: compactness, pseudocompactness, sigma-compactness, the property of being a Lindelof Sigma-space, the property of being a compact metrizable space, the (finite) number of connected components, connectedness, total disconnectedness. An example of R-equivalent (that is, l-equivalent) spaces that are not T-equivalent is constructed.