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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Sequential conditions and free topological groups

Authors: Edward T. Ordman and Barbara V. Smith-Thomas
Journal: Proc. Amer. Math. Soc. 79 (1980), 319-326
MSC: Primary 54D55; Secondary 22A99, 54G20
MathSciNet review: 565363
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Abstract: Most of the results in this paper concern relationships between sequential properties of a pointed topological space (X, p) and sequential properties of the Graev free topological group on X. In particular, it is shown that the free group over a sequential $ {k_\omega }$-space is sequential, and that a nondiscrete sequential free group has sequential order equal to $ {\omega _1}$ (the first uncountable ordinal). The free topological group on a space X which includes a convergent sequence contains a closed subspace homeomorphic to $ {S_\omega }$, a previously studied homogeneous, zero-dimensional sequential space. Finally, it is shown that there is no topological group homeomorphic to $ {S_\omega }$.

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Keywords: Sequential space, sequential order, $ {k_\omega }$-space, free topological group, Graev free topological group, sequential coreflection
Article copyright: © Copyright 1980 American Mathematical Society