Some theorems on colocally connected continua

TL;DR

This paper investigates properties of colocally connected continua, demonstrating how certain types of maps preserve or do not preserve colocal connectedness, and exploring its status as a Whitney property.

Contribution

It establishes that refinable maps preserve colocal connectedness, while proximately refinable maps do not, and clarifies the Whitney property status of colocal connectedness.

Findings

Refinable maps preserve colocal connectedness.

Proximately refinable maps may not preserve colocal connectedness.

Colocal connectedness is a Whitney property but not a Whitney reversible property.

Abstract

We show that each refinable map preserves colocal connectedness of the domain while a proximately refinable map does not necessarily. Also, we prove that colocal connectedness is a Whitney property and is not a Whitney reversible property.