Simply sm -factorizable (para)topological groups and their completions
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Simply sm-factorizable (para)topological groups and their completions Li-Hong Xie1 · Mikhail G. Tkachenko2 Received: 12 February 2020 / Accepted: 28 March 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract Let us call a (para)topological group strongly submetrizable if it admits a coarser separable metrizable (para)topological group topology. We present a characterization of simply sm-factorizable (para)topological groups by means of continuous real-valued functions. We show that a (para)topological group G is a simply sm-factorizable if and only if for each continuous function f : G → R, one can find a continuous homomorphism ϕ of G onto a strongly submetrizable (para)topological group H and a continuous function g : H → R such that f = g ◦ ϕ. This characterization is applied for the study of completions of simply sm-factorizable topological groups. We prove that the equalities μG = ω G = υG hold for each Hausdorff simply sm-factorizable topological group G, where υG and μG are the realcompactification and Dieudonné completion of G, respectively. This result gives a positive answer to a question posed by Arhangel’skii and Tkachenko in 2018. It is also proved that υG and μG coincide for every regular simply sm-factorizable paratopological group G and that υG admits the natural structure of paratopological group containing G as a dense subgroup and, furthermore, υG is simply sm-factorizable. Some results in [Completions of paratopological groups, Monatsh. Math. 183, 699–721 (2017)] are improved or generalized.
Communicated by John S. Wilson. Li-Hong Xie is supported by NSFC (Nos. 11601393; 11861018).
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Mikhail G. Tkachenko [email protected] Li-Hong Xie [email protected]; [email protected]
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School of Mathematics and Computational Science, Wuyi University, Jiangmen 529020, People’s Republic of China
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Departamento de Matemáticas, Universidad Autónoma Metropolitana, Av. San Rafael Atlixco 186, Col. Vicentina, Iztapalapa, CP 09340 Ciudad de México, Mexico
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L.-H. Xie, M. G. Tkachenko
Keywords Simply sm-factorizable · realcompactification · Dieudonné completion · Lindelöf -space · R-factorizable group Mathematics Subject Classification 22A05 · 22A30 · 54H11 · 54A25 · 54C30
1 Introduction A paratopological group G is a group G with a topology such that the multiplication mapping of G × G to G associating x y to arbitrary x, y ∈ G is jointly continuous. A paratopological group G is called a topological group if the inversion on G is continuous. Slightly reformulating the celebrated theorem of Comfort and Ross [5, Theorem 1.2], we can say that the pseudocompact topological groups are exactly the dense C-embedded subgroups of compact topological groups. In particular, the Stoneˇ Cech compactification, βG, the Hewitt-Nachbin completion, υG, and the Ra˘ıkov completion, G, of a pseudocompact topological group G coincide. Hence the Hewitt-Nachbin completion of the group G is again a topological group containing G as a dense C-embedded subgroup. Recently, with the idea to study c
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