Real-valued measurable cardinals and sequentially continuous homomorphisms

A. V. Arkhangel'skiĭ asked in 1981 if the variety V of topological groups generated by free topological groups on metrizable spaces coincides with the class of all topological groups. We show that if there exists a real-valued measurable cardinal then the variety V is a proper subclass of the class of all topological groups. A topological group G is called g-sequential if for any topological group H any sequentially continuous homomorphism G→H is continuous. We introduce the concept of a g-sequential cardinal and prove that a locally compact group is g-sequential if and only if its local weight is not a g-sequential cardinal. The product of a family of non-trivial g-sequential topological groups is g-sequential if and only if the cardinal of this family is not g-sequential. Suppose G is either the unitary group of a Hilbert space or the group of all self-homeomorphisms of a Tikhonov cube. Then G is g-sequential if and only if its weight is not a g-sequential cardinal. Every compact group of Ulam-measurable cardinality admits a strictly finer countably compact group topology.