Destruction of metrizability in generalized inverse limits (Q2800406)

Recently the concept of generalized inverse limits has been generalized to the case where the factor spaces are indexed by sets other than the positive integers. As the main result of the paper it is proved that if \(\kappa\) is a nonzero ordinal and if \(f: [0,1] \rightarrow C([0,1])\) is an upper semi-continuous, idempotent, and surjective map, but \(f\) is not the identity, then the inverse limit contains a copy of \(\kappa + 1\).NEWLINENEWLINEAs a corollary it follows that an inverse limit with a single upper semi-continuous, idempotent, surjective bonding function \(f: [0,1] \rightarrow C([0,1])\) and index set \(\kappa\), a nonzero ordinal, is only metric (in fact, Corson compact) in the case that \(\kappa\) is countable or the bonding map is trivial.