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We give an example as indicated in the title where the weight (i.e., the minimum cardinality of a base for the topology) of the product is the smallest uncountable cardinal. 0. INTRODUCTION AND THE MAIN THEOREM History. Throughout this paper the term "space" abbreviates "Hausdorff regular topological space". Is there a normal space and a metric space with a nonnormal product? The answer is yes in ZFC and is due to E. A. Michael ([6], 1963). (ZFC denotes the standard set of axioms for set theory.) Let IP and R denote respectively the irrationals and the reals with their usual topologies; and let M denote the Michael line which is the refinement of R obtained by isolating each irrational point. In Michael's example, M is the normal space (and moreover hereditarily paracompact) and P is the metric space. Note that the weight of M, and therefore the product, is c (c denotes the cardinality of the continuum of real numbers). Is there a ZFC example of a normal space with weight RI (the smallest uncountable cardinal) whose product with IP is nonnormal? (This question is due to Ronnie Levy.) It is not possible in ZFC to construct a Lindelbf space with both R, weight and a nonnormal product with IP (Lawrence, [5], 1990). Define a Michael space to be a Lindeldf space whose product with IP is nonnormal. The existence of a Michael space follows from both the Continuum Hypothesis (Michael, [7], 1971; first mentioned in a footnote of [6]) and Martin's Axiom (K. Alster, [1], 1990). Is there a Michael space in LZFC? This is Michael's problem and is equivalent to the existence in ZFC of a Lindeldf space and a separable completely metrizable space with a nonnormal product (see [5]). Suppose we drop the requirement for completeness on the metrizable factor. In the 1963 paper, Michael also gave a ZFC example of a Lindeldf space and a separable metric space with a nonnormal product. The Lindel6f space is the refinement of R obtained by isolating each point in a Bernstein set, and the metric space is the Bernstein set as a subspace of R. A subset of R is Bernstein if both it and its Received by the editors November 24, 1992 and, in revised form, March 14, 1994. 1991 Mathematics Subject Classification. Primary 54B10; Secondary 54D20, 54E50.

A ZFC example (of minimum weight) of a Lindelöf space and a completely metrizable space with a nonnormal product