A normal space X for which X×I is not normal

In [l ] C. Dowker gave a number of interesting characterizations of normal Hausdorff spaces whose cartesian product with the closed unit interval is not normal. Thus, such a space is often called a Dowker space; a Dowker space X will be described below. I t was previously known only that the existence of a Dowker space is consistent with the usual axioms of set theory [2], [3]. The proof that X is a Dowker space uses no set theoretic assumptions beyond the axiom of choice. We use the convention that an ordinal X is the set of all ordinals less than A. An ordinal y is said to be cofinal with X if there is a subset T of X order isomorphic with 7 such that a: (EX implies a ^/3 for some j3£T; let cf(X) be the smallest ordinal cofinal with X. Define F = {f:o)0—*o„| V^Eco0, f(n) ^a)n + l}. Define X= {fÇzF\ 3&Ek>0 such that VTZ£CO0, o)0<cï(f(n)) <œk}. For ƒ and g in Ft define Ufg= {hÇzX\ VwEco0, f(n) <h(n) Sg(n)}. Then topologize X by using the set of all U/g for ƒ and g in F as a basis for the topology. The resulting space is a collectionwise normal Dowker space.