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      Group-valued continuous functions with the topology of pointwise convergence

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          Abstract

          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.

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          On the existence of free topological groups

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            Free compact groups I: Free compact abelian groups

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                Author and article information

                Journal
                28 July 2009
                2010-04-23
                Article
                10.1016/j.topol.2009.06.022
                0907.4941

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                Custom metadata
                54C35 (Primary), 22A05 (Secondary), 46E10, 54H11
                Topology and its Applications, 157 (2010), 1518-1540
                Two references were added and one reference was updated. Question 11.1 was resolved in arXiv:0909.2381 [math.GN]. The bibliographic information related to Theorem 10.2 was corrected. Minor typos were corrected as well.
                math.GN math.FA math.GR

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