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## Archive of Issues

Russia Izhevsk
Year
2013
Issue
3
Pages
20-27
 Section Mathematics Title On compact $T_1$-spaces Author(-s) Voronov M.E.a Affiliations Udmurt State Universitya Abstract We consider spaces, any subspaces of which are compact. We call such spaces hereditarily compact. The present work covers questions on the existence and methods of constructing hereditarily compact $T_1$-topologies. We prove the existence of $2^{\tau}$ pairwise incomparable hereditarily compact $T_1$-topologies on an infinite set $X$ of power $\tau$. The characteristics of hereditarily compact spaces are obtained. It is proved that the Tychonoff product of a finite number of hereditarily compact $T_1$-spaces is a hereditarily compact $T_1$-space, but the Tychonoff product of an infinite number of nonsingleton hereditarily compact $T_1$-spaces is not hereditarily compact. Keywords compactness, minimal $T_1$-topology, Tychonoff product UDC 515.122.22, 515.122.252 MSC 54D10, 54D30 DOI 10.20537/vm130302 Received 22 July 2013 Language Russian Citation Voronov M.E. On compact $T_1$-spaces, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, issue 3, pp. 20-27. References de Groot Y. An isomorphism principle in general topology, Bull. Amer. Math. Soc., 1967, vol. 73, no. 3, pp. 465-467. Arkhangel'skii A.V. Maps and spaces, Usp. Mat. Nauk, 1966, vol. 21, no. 4, pp. 133-183. Arkhangel'skii A.V., Ponomarev V.I. Osnovy obshchei topologii v zadachakh i uprazhneniyakh (Fundamentals of general topology through problems and exercises), Moscow: Nauka, 1974, 424 p. Engelking R. Obshchaya topologiya (General topology), Moscow: Mir, 1986, 752 p. Fedorchuk V.V., Filippov V.V. Obshchaya topologiya. Osnovnye konstruktsii (General topology. Basic design), Moscow: Fizmatlit, 2006, 336 p. Full text