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

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