##
**Continuous homomorphic images to point-separating classes of topological groups.**
*(English)*
Zbl 1510.54020

A topological group \(G\) is said to be minimally almost periodic if each continuous homomorphism from \(G\) to a compact group is the trivial homomorphism. These topological groups were introduced by von Neumann, and have been at the center of a long list of high-profile problems in topological group theory. The following concept is introduced in this paper. A topological group \(G\) is said to be a \((\mathcal{C}\to\mathcal{D})\)-group if each continuous homomorphic image of \(G\) to an arbitrary group contained in the class \(\mathcal{C}\) is automatically contained in \(\mathcal{D}\) as well. The goal of this paper is to substantially simplify a way of characterizing a given class of \((\mathcal{C}\to\mathcal{D})\)-groups based on the underlying structure of the class \(\mathcal{C}\).

The main result of this paper aims to characterize all reflective classes \(\mathcal{C}\) for which \((\mathcal{C}\to\mathcal{D})\)-groups are verified exactly through explicit topological group quotients. The following theorem is proved. Let \(\mathcal{C}\) and \(\mathcal{D}\) be classes of topological groups. Assume \(\mathcal{C}\) is reflective and mono-transitive, and \(\mathcal{D}\) is closed under continuous homomorphisms. Then \(G\) is a \((\mathcal{C}\to\mathcal{D})\)-group if and only if, let \(K\) be a topological group quotient of \(G\), if \(K\) is contained in \(\mathcal{C}\), then it is also contained in \(\mathcal{D}\).

The main result of this paper aims to characterize all reflective classes \(\mathcal{C}\) for which \((\mathcal{C}\to\mathcal{D})\)-groups are verified exactly through explicit topological group quotients. The following theorem is proved. Let \(\mathcal{C}\) and \(\mathcal{D}\) be classes of topological groups. Assume \(\mathcal{C}\) is reflective and mono-transitive, and \(\mathcal{D}\) is closed under continuous homomorphisms. Then \(G\) is a \((\mathcal{C}\to\mathcal{D})\)-group if and only if, let \(K\) be a topological group quotient of \(G\), if \(K\) is contained in \(\mathcal{C}\), then it is also contained in \(\mathcal{D}\).

Reviewer: Shou Lin (Ningde)

### MSC:

54H11 | Topological groups (topological aspects) |

22Axx | Topological and differentiable algebraic systems |

54D05 | Connected and locally connected spaces (general aspects) |

### Keywords:

continuous homomorphic images; reflection of a topological group; topological group quotients; totally disconnected groups; maximally almost periodic groups
PDFBibTeX
XMLCite

\textit{V. H. Yañez}, Topology Appl. 326, Article ID 108418, 13 p. (2023; Zbl 1510.54020)

Full Text:
DOI

### References:

[1] | Arhangel’skiĭ, A.; Tkachenko, M., Topological Groups and Related Structures, Atlantis Stud. Math., vol. 1 (2008), Atlantis Press, World Scientific Publishing Co. Pte. Ltd.: Atlantis Press, World Scientific Publishing Co. Pte. Ltd. Paris, Hackensack, NJ, 781 pp · Zbl 1323.22001 |

[2] | Arhangel’skiĭ, A.; Wiegandt, R., Connectednesses and disconnectednesses in topology, Gen. Topol. Appl., 5, 9-33 (1975) · Zbl 0329.54008 |

[3] | Comfort, W. W.; Hernández, S.; Trigos-Arrieta, F. J., Relating a locally compact Abelian group to its Bohr compactification, Adv. Math., 120, 2, 322-344 (1996) · Zbl 0863.22004 |

[4] | Dikranjan, D.; Bruno, A. Giordano, w-Divisible groups, Topol. Appl., 155, 4, 252-272 (2008) · Zbl 1149.22004 |

[5] | Dikranjan, D.; Shakhmatov, D., A complete solution of Markov’s problem on connected group topologies, Adv. Math., 286, 286-307 (2016) · Zbl 1331.22003 |

[6] | Fuchs, L., Infinite Abelian Groups, Vol. I (1970), Academic Press: Academic Press New York · Zbl 0209.05503 |

[7] | Markov, A. A., On free topological groups, Izv. Ross. Akad. Nauk, Ser. Mat., 9, 3-64 (1945), (in Russian); English translation · Zbl 0061.04210 |

[8] | Markov, A. A., On unconditionally closed sets, (Topology and Topological Algebra. Topology and Topological Algebra, Transl. Ser. 1, vol. 8 (1962), AMS), 273-304 |

[9] | Preuss, G. (1967), Freie Universität: Freie Universität Berlin, Ph.D. Thesis |

[10] | Shakhmatov, D.; Spěvák, J., Group-valued continuous functions with the topology of pointwise convergence, Topol. Appl., 157, 1518-1540 (2010) · Zbl 1195.54040 |

[11] | von Neumann, J., Almost periodic functions in a group I, Trans. Am. Math. Soc., 36, 445-492 (1934) · Zbl 0009.34902 |

[12] | von Neumann, J.; Wigner, E., Minimally almost periodic groups, Ann. Math., 41, 746-750 (1940) |

[13] | Yañez, V. H., Topological groups described by their continuous homomorphisms or small subgroups, (The 67th Topology Symposium (Abstract) (2020), Japanese Mathematical Society), 10 pages. Open access: |

[14] | Yañez, V. H., Properties modelled on minimal almost periodicity, and small subgroup generating properties (2021), Ehime University: Ehime University Matsuyama, Japan, PhD Thesis |

[15] | Yañez, V. H., Topological groups without infinite precompact continuous homomorphic images, Topol. Appl., 301, Article 107544 pp. (2021) · Zbl 1479.22004 |

[16] | Yañez, V. H., Group topologies making every continuous homomorphic image to a compact group connected, Topol. Appl., 311, Article 107958 pp. (2022) · Zbl 1496.54021 |

[17] | V.H. Yañez, Strengthening minimal almost periodicity via the classical triad resolving Hilbert’s Fifth problem, in preparation. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.