Connected economically metrizable spaces

Taras Banakh¹ ; Myroslava Vovk² ; Michał Ryszard Wójcik³

¹ Instytut Matematyki Uniwersytet Humanistyczno-Przyrodniczy Jana Kochanowskiego Kielce, Poland and Department of Mathematics Ivan Franko National University of Lviv Universytetska 1 79000, Lviv, Ukraine
² National University Lvivska Politechnika Lviv, Ukraine
³ Department of Mathematics University of Louisville Louisville, KY, U.S.A. and Institute of Geography and Regional Development University of Wrocław 50-137 Wrocław, Poland

Fundamenta Mathematicae, Tome 212 (2011) no. 2, pp. 145-173

Résumé

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.

Détail

DOI : 10.4064/fm212-2-3

Keywords: topological space non separably connected connected its connected separable subspaces singletons each connected sequential topological space image non separably connected complete metric space cal under monotone quotient map metric cal space cal economical sense each infinite subspace subset cardinality set cal does exceed density cal times dens construction space cal determines functor cal top metr category top topological spaces their continuous maps category metr metric spaces their non expanding maps