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## Archive of Issues

Russia Izhevsk
Year
2018
Volume
28
Issue
3
Pages
277-283
 Section Mathematics Title On convergent sequences and properties of subspaces Author(-s) Gryzlov A.A.a, Tsigvintseva K.N.a Affiliations Udmurt State Universitya Abstract We consider problems connected with the notion of convergent sequences in $T_1$-spaces. The properties of $T_1$-spaces and convergent sequences in these spaces considerably differ from the same properties of Hausdorff spaces. We consider problems connected with the properties of the minimal $T_1$-space. We consider properties of spaces where every sequence is a convergent sequence (Theorems 1 and 2 and Example 1). One of the main problems is the connection between convergent sequences and the properties of subspaces of the space. It is well known that the compactness, countable compactness and sequential compactness are not equivalent in general. We prove (Theorem 7) that hereditary sequential compactness, compactness and countable compactness are equivalent. Keywords convergent sequence, $T_1$-compactness, compactness UDC 515.122.22, 515.122.252 MSC 54D10, 54D30 DOI 10.20537/vm180301 Received 25 July 2018 Language Russian Citation Gryzlov A.A., Tsigvintseva K.N. On convergent sequences and properties of subspaces, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 3, pp. 277-283. References Aleksandrov P.S. Vvedenie v teoriyu mnozhestv i obshchuyu topologiyu (Introduction to set theory and general topology), Moscow: Nauka, 1977, 368 p. Arkhangel'skii A.V., Ponomarev V.I. Osnovy obshchei topologii v uprazhneniyakh i zadachakh (Fundamentals of general topology through problems and exercises), Moscow: Nauka, 1974, 423 p. Arkhangel'skii A.V. Mappings and spaces, Russian Math. Surveys, 1966, vol. 21, no. 4, pp. 115-162. DOI: 10.1070/RM1966v021n04ABEH004169 Bonanzinga M., Stavrova D., Staynova P. Combinatorial separation axioms and cardinal invariants, Topology and its Applications, 2016, vol. 201, pp. 441-451. DOI: 10.1016/j.topol.2015.12.053 Voronov M.E. On compact $T_1$-space, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, issue 3, pp. 20-27 (in Russian). DOI: 10.20537/vm130302 Engelking R. General topology, Warszawa: PWN - Polish Scientific Publishers, 1977, 626 p. Translated under the title Obshchaya topologia, Moscow: Nauka, 1986, 752 p. Gryzlov A.A. Two theorems on the cardinality of topological spaces, Dokl. Akad. Nauk SSSR, 1980, vol. 251, no. 4, pp. 780-783 (in Russian). Gryzlov A. On some cardinal invariants of compact and $H$-closed spaces, ItEs-2007: Abstracts of Sixth Italian-Spanish conf. on General Topology and Applications, Bressanone, Italia, 2007, p. 47. Reilly I.L. On non-Hausdorff spaces, Topology and its Applications, 1992, vol. 44, issues 1-3, pp. 331-340. DOI: 10.1016/0166-8641(92)90106-A Full text