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

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.