Connected economically metrizable spaces
Volume 212 / 2011
Abstract
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.
