Group-valued continuous functions with the topology of pointwise convergence - Mathematics > General TopologyReport as inadecuate

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Abstract: We denote by C pX,G the group of all continuous functions from a space X toa topological group G endowed with the topology of pointwise convergence. Wesay that spaces X and Y are G-equivalent provided that the topological groupsC pX,G and C pY,G are topologically isomorphic. We investigate whichtopological properties are preserved by G-equivalence, with a special emphasisbeing placed on characterizing topological properties of X in terms of those ofC pX,G. Since R-equivalence coincides with l-equivalence, this line ofresearch -includes- major topics of the classical C p-theory of Arhangel-skiias a particular case when G = R. We introduce a new class of TAP groups thatcontains all groups having no small subgroups NSS groups. We prove that: ifor a given NSS group G, a G-regular space X is pseudocompact if and only ifC pX,G is TAP, and ii for a metrizable NSS group G, a G^*-regular space Xis compact if and only if C pX,G is a TAP group of countable tightness. Inparticular, a Tychonoff space X is pseudocompact compact if and only ifC pX,R is a TAP group of countable tightness. We show that Tychonoff spacesX and Y are T-equivalent if and only if their free precompact Abelian groupsare topologically isomorphic, where T stays for the quotient group R-Z. As acorollary, we obtain that T-equivalence implies G-equivalence for every Abelianprecompact group G. We establish that T-equivalence preserves the followingtopological properties: compactness, pseudocompactness, sigma-compactness, theproperty of being a Lindelof Sigma-space, the property of being a compactmetrizable space, the finite number of connected components, connectedness,total disconnectedness. An example of R-equivalent that is, l-equivalentspaces that are not T-equivalent is constructed.

Author: Dmitri Shakhmatov, Jan Spěvák



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