The Cantor tree and the Fréchet-Urysohn property. (English) Zbl 0894.54021

Kopperman, Ralph (ed.) et al., Papers on general topology and related category theory and topological algebra. Proceedings of the 2nd and 3rd conferences on limits, New York, NY, USA, July 2–3, 1985 and June 12–13, 1987. New York, NY: New York Academy of Sciences. Ann. N. Y. Acad. Sci. 552, 109-123 (1989).
This paper is one of the earlier and more important contributions to the investigation of countable Fréchet-Urysohn \(\alpha_i\)-spaces (Definition 2.3). The author begins with the well-known Cantor tree (namely \(2^{\leq \omega}\) with the tree topology) and performs two simple operations to generate a variety of Fréchet-Urysohn spaces. For each \(A\subset 2^\omega\) and \(T=2^{<\omega}\), \(T\cup A\) is locally compact and the one-point compactification, denoted \(T\cup A + \infty \) in the paper, is shown to be Fréchet-Uryson and, for uncountable \(A\), not first countable. The most important result is that if \(A\) is an uncountable \(\lambda'\)-set (which exists in ZFC [see A. Miller, Handbook of set-theoretic topology, 201-233 (1984; Zbl 0588.54035)]) then this space is an \(\alpha_2\)-space (or \(w\)-space). In addition, it is established that \(A\) being a \(\lambda'\)-set is squeezed somewhere between \(T\cup A+\infty\) being an \(\alpha_1\)-space and its being an \(\alpha_2\)-space. The problem is posed as to whether the latter is an equivalence. Finally, it is shown that \(T\cup A+\infty\) naturally generates a countable topological group which is Fréchet-Urysohn if (and only if) \(A\) is a \(\gamma\)-set. This, we believe was the first (consistent) example of a countable Fréchet-Urysohn non-metrizable topological group. J. Gerlits and Zs. Nagy introduced \(\gamma\)-sets [Topology Appl. 14, 151-161 (1982; Zbl 0503.54020)]. Some readers might enjoy the elegant realization of the Cantor tree as a closed subset of the Sorgenfrey plane.
For the entire collection see [Zbl 0879.00050].
Reviewer: A.Dow (North York)


54D55 Sequential spaces
54H11 Topological groups (topological aspects)