Mahavier Products, Idempotent Relations, and Condition $\Gamma$

TL;DR

This paper explores conditions under which Mahavier Products and inverse limits of metrizable spaces are themselves metrizable, revealing that weaker criteria than previously thought suffice for metrizability.

Contribution

The authors establish a more general metrizability theorem for Mahavier Products, showing that Condition $\Gamma$ is guaranteed by weaker criteria than previously known.

Findings

Generalized inverse limits of metrizable spaces are metrizable.

Condition $\Gamma$ is satisfied under weaker conditions.

Metrizability of Mahavier Products is characterized by these weaker criteria.

Abstract

Clearly, a generalized inverse limit of metrizable spaces indexed by $\mathbb N$ is metrizable, as it is a subspace of a countable product of metrizable spaces. The authors previously showed that all idempotent, upper semi-continuous, surjective, continuum-valued bonding functions on $[0,1]$ (besides the identity) satisfy a certain Condition $\Gamma$; it follows that only in trivial cases can a generalized inverse limit of copies of ([0,1]) indexed by an uncountable ordinal be metrizable. The authors show that Condition $\Gamma$ is in fact guaranteed by much weaker criteria, proving a more general metrizability theorem for certain Mahavier Products.