Connected economically metrizable spaces

Tom 212 / 2011

Taras Banakh, Myroslava Vovk, Michał Ryszard Wójcik Fundamenta Mathematicae 212 (2011), 145-173 MSC: Primary 54B30, 54D05, 54F15; Secondary 54C30, 54E35, 54E50, 54G20, 54G15. DOI: 10.4064/fm212-2-3


A topological space is non-separably connected if it is connected but all of its connected separable subspaces are singletons. We show that each connected sequential topological space $X$ is the image of a non-separably connected complete metric space ${\cal E} X$ under a monotone quotient map. The metric $d_{{\cal E} X}$ of the space ${\cal E} X$ is economical in the sense that for each infinite subspace $A\subset X$ the cardinality of the set $\{d_{{\cal E} X}(a,b):a,b\in A\}$ does not exceed the density of $A$, $|d_{{\cal E} X}(A\times A)|\le{\rm dens}(A)$.

The construction of the space ${\cal E} X$ determines a functor ${\cal E}:{\rm Top}\to{\rm Metr}$ from the category ${\rm Top}$ of topological spaces and their continuous maps into the category ${\rm Metr}$ of metric spaces and their non-expanding maps.


  • Taras BanakhInstytut Matematyki
    Uniwersytet Humanistyczno-Przyrodniczy
    Jana Kochanowskiego
    Kielce, Poland
    Department of Mathematics
    Ivan Franko National University of Lviv
    Universytetska 1
    79000, Lviv, Ukraine
  • Myroslava VovkNational University
    Lvivska Politechnika
    Lviv, Ukraine
  • Michał Ryszard WójcikDepartment of Mathematics
    University of Louisville
    Louisville, KY, U.S.A.
    Institute of Geography and Regional Development
    University of Wrocław
    50-137 Wrocław, Poland

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