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Relations of topologies as tools of (bi)topological applications

  • Badri P. Dvalishvili EMAIL logo
Published/Copyright: November 10, 2015
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Georgian Mathematical Journal
From the journal Georgian Mathematical Journal Volume 22 Issue 4

Abstract

For a bitopological space (X,τ1<τ1(A)), where A∉τ1 and τ1(A) is a simple extension of the initial topology τ1, we show under which relations between the topologies τ1 and τ1(A) such properties as in particular the Blumberg property, pseudocompleteness, connectedness, submaximality, to be a Baire space, to be an H-closed space, to be an HD-space, to be an eD-space and to be a nodec space are passed from τ1 over to τ1(A). Moreover, we give the necessary and sufficient conditions both for the connectedness of a simple extension and for the maximal connectedness of some special topology. We obtain a solution for one of Everett and Ulam's problems dealing with the case of coincidence of two classes of homeomorphisms of a space onto itself.

Keywords: S-relation; coupling of topologies; nearness of topologies; simple extension; adjacent topologies; weak Lindelöf number; HD-space; eD-space; (i,j)-submaximal space; (i,j)-singular set
MSC: 54A10; 54E55
Received: 2015-7-15
Accepted: 2015-8-27
Published Online: 2015-11-10
Published in Print: 2015-12-1

© 2016 by De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. Algorithmic problems for class-2 nilpotents MR-groups
  3. Morava K(s)*-rings of the extensions of Cp by the products of good groups under diagonal action
  4. Singular homology theory in the category of soft topological spaces
  5. S4.3 and hereditarily extremally disconnected spaces
  6. Relations of topologies as tools of (bi)topological applications
  7. The twisted Cartesian model for the double path fibration
  8. Removable sets for harmonic functions in Besov spaces
  9. Racks, Leibniz algebras and Yetter–Drinfel'd modules
  10. Homotopy properties of differential modules with simplicial F∞-faces and D∞-differential modules
  11. The fundamental groupoid as a terminal costack
  12. Relative topological finiteness
https://doi.org/10.1515/gmj-2015-0047
Keywords for this article
S-relation; coupling of topologies; nearness of topologies; simple extension; adjacent topologies; weak Lindelöf number; HD-space; eD-space; (i,j)-submaximal space; (i,j)-singular set

Articles in the same Issue

  1. Frontmatter
  2. Algorithmic problems for class-2 nilpotents MR-groups
  3. Morava K(s)*-rings of the extensions of Cp by the products of good groups under diagonal action
  4. Singular homology theory in the category of soft topological spaces
  5. S4.3 and hereditarily extremally disconnected spaces
  6. Relations of topologies as tools of (bi)topological applications
  7. The twisted Cartesian model for the double path fibration
  8. Removable sets for harmonic functions in Besov spaces
  9. Racks, Leibniz algebras and Yetter–Drinfel'd modules
  10. Homotopy properties of differential modules with simplicial F∞-faces and D∞-differential modules
  11. The fundamental groupoid as a terminal costack
  12. Relative topological finiteness
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