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## Archive of Issues

Russia Moscow; Yekaterinburg
Year
2015
Volume
25
Issue
2
Pages
180-183
 Section Mathematics Title Example of nonnormal submetrizable space of small cardinality Author(-s) Patrakeev M.A.ab, Reznichenko E.A.c Affiliations Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa, Ural Federal Universityb, Lomonosov Moscow State Universityc Abstract In 1976 K. Alster and T. Przymusinski constructed an example of nonnormal completely regular separable and first-countable topological space of cardinality $\aleph_1$. Also they proved a theorem which implies that under Martin's Axiom with negation of continuum hypothesis there is no similar example, which is additionally cometrizable. If we weaken the cometrizability condition to submetrizability, then the similar statement can not be proved: here we construct an example of nonnormal completely regular submetrizable separable first-countable locally countable space of cardinality $\aleph_1.$ Keywords submetrizable space, nonnormal space, cometrizable space UDC 515.122 MSC 54G20, 54D15 DOI 10.20537/vm150203 Received 19 May 2015 Language Russian Citation Patrakeev M.A., Reznichenko E.A. Example of nonnormal submetrizable space of small cardinality, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 2, pp. 180-183. References Alster K., Przymusinski T. Normality and Martin's axiom, Fund. Math., 1976, vol. 91, pp. 123-131. Barwise J. (Ed.) Handbook of mathematical logic, Amsterdam: Elsevier, 1977, 1165 p. Kunen K., Vaughan J.E. (Ed.) Handbook of set-theoretic topology, Amsterdam: North-Holland, 1984, 1273 p. Engelking R. General topology, Sigma series in pure mathematics, vol. 6, Berlin: Heldermann-Verlag, 1989, 529 p. Gartside P.M., Reznichenko E.A., Sipacheva O.V. Mal'tsev and retral spaces, Topology Appl., 1997, vol. 80, no. 1-2, pp. 115-129. Full text