Connected, not separably connected complete metric spaces

TL;DR

This paper introduces a method to construct complete connected metric spaces with specific properties, such as having zero-dimensional separable subsets, by transforming and taking inverse limits of bounded metric spaces.

Contribution

It presents a novel mechanism for building complete connected metric spaces with prescribed separablewise component structures, expanding understanding of space construction techniques.

Findings

Constructed a complete connected metric space with zero-dimensional separable subsets.

Developed a mechanism to produce spaces with quotient structures isometric to original spaces.

Demonstrated the inverse limit construction for complex connected spaces.

Abstract

In a separably connected space any two points are contained in a separable connected subset. We show a mechanism that takes a connected bounded metric space and produces a complete connected metric space whose separablewise components form a quotient space isometric to the original space. We repeatedly apply this mechanism to construct, as an inverse limit, a complete connected metric space whose each separable subset is zero-dimensional.