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• W2026513647 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 ku-space is sequential, and that a nondiscrete sequential free group has sequential order equal to w1 (the first uncountable ordinal). The free topological group on a space X which includes a convergent sequence contains a closed subspace homeomorphic to S,, a previously studied homogeneous, zero-dimensional sequential space. Finally, it is shown that there is no topological group homeomorphic to S.. 0. Introduction. In this paper we discuss relationships between sequential properties of a pointed topological space (X, p) and sequential properties of its Graev free topological group FG(X, p). Sequential spaces have been discussed in [D], [Frl], [Fr2], [A-Fr], [RI; we make heavy use of the space S. of sequential order co' constructed in [A-Fr]. The Graev free topological group FG(X, p) on a pointed Tychonoff space (X, p) was introduced in [Grl]; its topology has proved to be rather intractable, but in the last few years good results have been obtained in the case when X is a k.-space, that is, a weak union of countably many compact subsets [O1], [H-M], [MMO]. Definitions and preliminary results about sequential spaces appear in ?1; ?2 contains preliminaries about free topological groups and about k.-spaces. ?3 contains results about sequential properties of free topological groups and their consequences. The result that S, supports no group structure (answering in the negative a question of S. P. Franklin) appears in ?4. We thank S. P. Franklin for posing to us the question just mentioned, and for several helpful conversations and suggestions. 1. Sequential spaces. Our definition of sequential spaces, sequential order, and the particular example S.,, are based on [A-Fr]. A subset U of a topological space X is sequentially open if each sequence converging to a point in U is eventually in U. The space X is sequential if each sequentially open subset of X is open. For each subset A of X, let s(A) denote the set of all limits of sequences of points of A. X is of sequential order 1 (X is also called a Frh received by the editors February 5, 1979 and, in revised form, June 4, 1979. AMS (MOS) subject classifications (1970). Primary 54D55, 22A99, 20E05." @default.
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• W2026513647 date "1980-02-01" @default.
• W2026513647 modified "2023-10-17" @default.
• W2026513647 title "Sequential conditions and free topological groups" @default.
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• W2026513647 doi "https://doi.org/10.1090/s0002-9939-1980-0565363-2" @default.
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