PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 103, Number 2, June 1988 SEQUENTIAL CONDITIONS AND FREE PRODUCTS OF TOPOLOGICAL GROUPS SIDNEY A. MORRIS AND H. B. THOMPSON (Communicated by Dennis Burke) ABSTRACT. If A and B are nontrivial topological groups, not both discrete, such that their free product A 11 B is a sequential space, then it is sequential of order oji. 1. Introduction. In [15], Ordman and Smith-Thomas prove that if the free topological group on a nondiscrete space is sequential then it is sequential of order uji. In particular, this implies that free topological groups are not metrizable or even Fréchet spaces. Our main result is the analogue of this for free products of topological groups. More precisely we prove that if A and B are nontrivial topological groups not both discrete and their free product A II B is sequential, then it is sequential of order ui. This result is then extended to some amalgamated free products. Our theorem includes, as a special case, the result of [10]. En route we extend the Ordman and Smith-Thomas result to a number of other topologies on a free group including the Graev topology which is the finest locally invariant group topology. (See [12].) It should be mentioned, also, that O.dman and Smith- Thomas show that the condition of AII B being sequential is satisfied whenever A and B are sequential fc^-spaces. 2. Preliminaries. The following definitions and examples are based on Franklin [3, 4] and Engelking [2]. DEFINITIONS. A subset U of a topological space X is said to be sequentially open if each sequence converging to a point in U is eventually in U. The space X is said to be sequential if each sequentially open subset of X is open. REMARKS. A closed subspace of a sequential space is sequential. A subspace of a sequential space need not be sequential. (See Example 1.8 of [3].) DEFINITIONS. For each subset A of a sequential space X, let s (A) denote the set of all limits of sequences of points of A. The space X is said to be sequential of order 1 if s(A) is the closure of A for every A. The higher sequential orders are defined by induction. Let so(A) = A, and for each ordinal a = ß + 1, let sa(A) = s(sß(A)). If a is a limit ordinal, let sa(A) = (J{sß(A): ß < a}. The sequential order of X is defined to be the least ordinal a such that sa(A) is the closure of A for every subset A of X. REMARKS. The sequential order always exists and does not exceed the first uncountable ordinal wi. Sequential spaces of order 1 are also known as Fréchet Received by the editors March 14, 1986 and, in revised form, February 6, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 22A05, 54D55, 20E06. Key words and phrases. Sequential space, sequential order, free product of topological groups, Graev topology, free abelian topological group. ©1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page 633 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use