Connected economically metrizable spaces

TL;DR

This paper constructs a functor that assigns to each topological space a complete metric space with economical metric properties, illustrating a deep connection between topological and metric structures.

Contribution

It introduces a novel functor Eco that maps topological spaces to complete, economically metrizable spaces, expanding understanding of nonseparably connected spaces.

Findings

Eco(X) is a complete, economical metric space.

Eco(X) maps onto any connected sequential space X.

The construction defines a functor from Top to Metr categories.

Abstract

A topological space is nonseparably 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 nonseparably connected complete metric space Eco(X) under a monotone quotient map. The metric d of the space Eco(X) is economical in the sense that for each infinite subspace A of X the cardinality of the set {d(a,b):a,b in A} does not exceed the density of A. The construction of the space Eco(X) determines a functor Eco from the category Top of topological spaces and their continuous maps into the category Metr of metric spaces and their non-expanding maps.