2009/07/31 by Dmitri Shakhmatov, Jan Spěvák · 1 citation
Mathematics · #Abelian group #Advanced Topology and Set Theory #Combinatorics #Countable set #Discrete mathematics #Group (periodic table) #Group action #Homotopy and Cohomology in Algebraic Topology #Locally compact space #Mathematical analysis #Mathematics #Metrization theorem #Network topology #Physics #Pointwise #Pointwise convergence #Pure mathematics #Quotient #Quotient space (topology) #Rings, Modules, and Algebras #Separable space #Topological group #Topological space #Topology (electrical circuits) #Tychonoff space #math.FA #math.GN #math.GR #msc:22A05 #msc:46E10 #msc:54C35 #msc:54H11
paper · pdf · doi:10.1016/j.topol.2009.06.022
published as 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.
openalex publication_date 2009/08/27 · crossref created 2009/08/27 · arxiv created 2010/04/23 · arxiv updated 2010/04/26 · crossref issued 2010/06/01 · crossref published 2010/06/01 · crossref published-print 2010/06/01 · openalex created_date 2016/06/24 · crossref deposited 2019/05/22 · crossref indexed 2026/08/04 · openalex updated_date 2026/08/05
We denote by Cp(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 Cp(X,G) and Cp(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 Cp(X,G). Since R-equivalence coincides with l-equivalence, this line of research "includes" major topics of the classical Cp-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 Cp(X,G) is TAP, and (ii) for a metrizable NSS group G, a G^*-regular space X is compact if and only if Cp(X,G) is a TAP group of countable tightness. In particular, a Tychonoff space X is pseudocompact (compact) if and only if Cp(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.