Nonseparably connected complete metric spaces

TL;DR

This paper constructs complete metric spaces that are connected but have only singleton separable connected subspaces, showing a new way to generate such spaces from first countable connected spaces.

Contribution

It demonstrates that every connected first countable space can be obtained as a continuous image of a nonseparably connected complete metric space.

Findings

Every connected first countable space is an image of a nonseparably connected complete metric space.

Introduces a method to construct nonseparably connected complete metric spaces.

Provides insight into the structure of connected metric spaces with trivial separable subspaces.

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 first countable space is the image of a nonseparably connected complete metric space under a continuous monotone hereditarily quotient map.