2021/08/22 by Vladimir Uspenskij, Uspenskij, Vladimir
Decision Sciences · Mathematics · #03E55 #22B05 #54E35 #Advanced Topology and Set Theory #FOS: Mathematics #Fuzzy and Soft Set Theory #General Topology (math.GN) #Homotopy and Cohomology in Algebraic Topology #Primary 22A05 #secondary 54C08
paper · pdf · doi:10.48550/arxiv.2108.09839
openalex publication_date 2021/08/22 · openalex created_date 2021/08/30 · openalex updated_date 2026/07/28
A.V.Arkhangel'skii asked in 1981 if the variety \mathfrak 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 \mathfrak 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.