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Section
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Mathematics
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Title
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Example of nonnormal submetrizable space of small cardinality
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Author(-s)
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Patrakeev M.A.ab,
Reznichenko E.A.c
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Affiliations
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Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesa,
Ural Federal Universityb,
Lomonosov Moscow State Universityc
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Abstract
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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 $\aleph_1.$
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Keywords
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submetrizable space, nonnormal space, cometrizable space
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UDC
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515.122
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MSC
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54G20, 54D15
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DOI
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10.20537/vm150203
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Received
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19 May 2015
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Language
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Russian
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Citation
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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.
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References
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- 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.
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Full text
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