One Dimensional Locally Connected S-spaces

TL;DR

This paper constructs a one-dimensional, locally connected hereditarily separable continuum without convergent sequences using inverse limits and the Menger sponge, assuming Jensen's diamond principle.

Contribution

It introduces a novel construction of a one-dimensional S-space with specific topological properties, extending Fedorchuk's original approach.

Findings

Constructed a one-dimensional hereditarily separable continuum without convergent sequences.

Used inverse limit of omega_1 steps with Menger sponge spaces.

Relied on Jensen's diamond principle for the construction.

Abstract

We construct, assuming Jensen's principle diamond, a one-dimensional locally connected hereditarily separable continuum without convergent sequences. The construction is an inverse limit in omega_1 steps, and is patterned after the original Fedorchuk construction of a compact S-space. To make it one-dimensional, each space in the inverse limit is a copy of the Menger sponge.