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Archive of Issues

Russia Izhevsk
Year
2019
Volume
29
Issue
3
Pages
312-318
 Section Mathematics Title On tightness and pseudocharacter of compact $T_1$-spaces Author(-s) Gryzlov A.A.a, Golovastov R.A.a Affiliations Udmurt State Universitya Abstract We consider the relationship between the pseudocharacter $\psi(X)$ and the tightness $t(X)$ of compact $T_1$-spaces $X$. We prove that $t(X)\leqslant\psi(X)$ for self-adjoined $T_1$-spaces, i.e., the spaces where a subset is closed if and only if it is compact. We also show that in general for compact $T_1$-spaces there is no relationship between these cardinal invariants. We give an example of a compact $T_1$-space such that the tightness of this space is uncountable, but its pseudocharacter is countable. Another example shows the space $X$ whose tightness is countable, but its pseudocharacter is uncountable. Keywords $T_1$-space, compact, tightness, pseudocharacter UDC 515.122.22, 515.122.252 MSC 54D10, 54D30 DOI 10.20537/vm190302 Received 15 July 2019 Language Russian Citation Gryzlov A.A., Golovastov R.A. On tightness and pseudocharacter of compact $T_1$-spaces, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 3, pp. 312-318. References Aleksandrov P.S., Uryson P.S. Memuar o kompaktnykh topologicheskikh prostranstvakh (A memoir on compact topological spaces), Moscow: Nauka, 1971, 144 p. Aleksandrov P.S. Vvedenie v teoriyu mnozhestv i obshchuyu topologiyu (Introduction to set theory and general topology), Moscow: Nauka, 1977, 368 p. Engelking R. General topology, Warszawa: PWN - Polish Scientific Publishers, 1977, 626 p. Translated under the title Obshchayua topologiya, Moscow: Nauka, 1986, 752 p. Arkhangelskii A.V. Mappings and spaces, Russian Mathematical Surveys, 1966, vol. 21, no. 4, pp. 115-162. https://doi.org/10.1070/RM1966v021n04ABEH004169 Arkhangelskii A.V. Structure and classification of topological spaces and cardinal invariants, Russian Mathematical Surveys, 1978, vol. 33, no. 6, pp. 33-96. https://doi.org/10.1070/RM1978v033n06ABEH003884 Gryzlov A.A. Two theorems on the cardinality of topological spaces, Soviet Math. Dokl., 1980, vol. 21, pp. 506-509. Voronov M.E. On compact $T_1$-spaces, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, issue 3, pp. 20-27 (in Russian). https://doi.org/10.20537/vm130302 Gryzlov A.A., Tsigvintseva K.N. On convergent sequences and properties of spaces, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 3, pp. 277-283 (in Russian). https://doi.org/10.20537/vm180301 Gryzlov A.A. On the properties of subsets of Tychonoff products, Topology and its Applications, 2016, vol. 201, pp. 13-17. https://doi.org/10.1016/j.topol.2015.12.023 Gryzlov A.A On dense subsets of Tychonoff products of $T_1$-spaces, Topology and its Applications, 2018, vol. 248, pp. 164-175. https://doi.org/10.1016/j.topol.2018.09.003 Gotchev I.S. Generalizations of two cardinal inequalities of Hajnal and Juhász, Topology and its Applications, 2017, vol. 221, pp. 425-431. https://doi.org/10.1016/j.topol.2017.02.026 Juhász I., Soukup L., Szentmiklóssy Z. First countable almost discretely Lindelöf $T_3$ spaces have cardinality at most continuum, Topology and its Applications, 2018, vol. 241, pp. 145-149. https://doi.org/10.1016/j.topol.2018.03.026 Carlson N.A., Porter J.R. On the cardinality of Hausdorff spaces and H-closed spaces, Topology and its Applications, 2018, vol. 241, pp. 377-395. https://doi.org/10.1016/j.topol.2017.01.027 Bonanzinga M., Stavrova D., Staynova P. Separation and cardinality - Some new results and old questions, Topology and its Applications, 2017, vol. 221, pp. 556-569. https://doi.org/10.1016/j.topol.2017.02.007 Full text