Workshop Lecture on Products of Fréchet Spaces

The general question, “When is the product of Fréchet spaces Fréchet?” really depends on the questions of when a product of α4 Fréchet spaces (also known as strongly Fréchet or countably bisequential spaces) is α4, and when it is Fréchet. Two subclasses of the class of strongly Fréchet spaces shed much light on these questions. These are the class of α3 Fréchet spaces and its subclass of א0-bisequential spaces. The latter is closed under countable products, the former not even under finite products. A number of fundamental results and open problems are recalled, some further highlighting the difference between being α3 and Fréchet and being א0-bisequential. This paper is a slightly updated note for the first of two lectures presented by the author in his Workshop on Sequential Convergence at the ten-day Advances in Set-Theoretic Topology conference in Erice. Erice is a remarkable mountaintop town with a medieval feel to it, overlooking the northwestern tip of Sicily, and the author is grateful for the invitation to this unique conference. Recall that a space is called Fréchet (or: Fréchet-Urysohn) if, whenever a point p is in the closure of a subset A, there is a sequence in A converging to p. In this paper, “space” will mean “Hausdorff space,” although much of what we say holds for topological spaces in general. This paper revolves around the following general problem, to which Tsugunori Nogura has made many basic contributions. General Problem. When is the product of Fréchet spaces Fréchet? 1. Fundamental problems and theorems The following space is very relevant to this general problem. Typeset by AMS-TEX 1