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      Selective sequential pseudocompactness

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          Abstract

          We say that a topological space X is selectively sequentially pseudocompact (SSP for short) if for every sequence (U_n) of non-empty open subsets of X, one can choose a point x_n in U_n for every n in such a way that the sequence (x_n) has a convergent subsequence. We show that the class of SSP spaces is closed under taking arbitrary products and continuous images, contains the class of all dyadic spaces and forms a proper subclass of the class of strongly pseudocompact spaces introduced recently by Garc\'ia-Ferreira and Ortiz-Castillo. We investigate basic properties of this new class and its relations with known compactness properties. We prove that every omega-bounded (=the closure of which countable set is compact) group is SSP, while compact spaces need not be SSP. Finally, we construct SSP group topologies on both the free group and the free Abelian group with continuum-many generators.

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          Zero-dimensionality of some pseudocompact groups

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            A pseudocompact Tychonoff space all countable subsets of which are closed and C∗-embedded

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              Group-valued continuous functions with the topology of pointwise convergence

              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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                Author and article information

                Journal
                2017-02-07
                Article
                1702.02055
                6a7c63f2-0dfb-4daa-96a6-c02d3582ec20

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

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                Custom metadata
                54D30 (Primary), 22A05, 22C05, 54A20, 54H11 (Secondary)
                math.GN

                Geometry & Topology
                Geometry & Topology

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