In 1977, the second author announced the following consistent neg- ative answer to a question of Katetov: Assuming MA+-GH , there is a compact nonmetric space X such that X1 is hereditarily normal. We give the details of this example, and construct another example assuming CH . We show that both examples can be constructed so that X2\A is perfectly normal. We also construct in ZFC a compact nonperfectly normal X such that X2\A is nor- mal. In his classical paper (K), Katetov showed that if X and Y are infinite compact spaces and X x Y is hereditarily normal, then X and Y are per- fectly normal. By Sneider's theorem that a compact space with a GVdiagonal is metrizable (S), Katetov concludes that if X is compact and X3 is heredi- tarily normal, then X is metrizable. He asked if the same conclusion could be obtained assuming only that X2 is hereditarily normal. In 1977, the second au- thor obtained a counterexample assuming Martin's Axiom plus the negation of the Continuum Hypothesis (MA + -CH). This result was announced in (Ny,), and an outline of the proof appeared there, although with many details omitted. A complete proof appears in this paper for the first time. We also construct a (necessarily different, as will be seen) counterexample assuming CH. Since any counterexample must be perfectly normal, it is probably not sur- prising that our examples are related to Alexandrov's double arrow space D = (0, 1) x {0, 1} with the lexicographic order topology, for it is in some sense the only known example in ZFC of a compact perfectly normal nonmetrizable space. (See (Gi) for a discussion of this.) The double arrow space has also been called the split interval because one can think of obtaining it by splitting each x € (0, 1) into two points x~ and x+ , and putting the order topology on these points, where x~ is declared to be less than x+ and otherwise the order is the natural one inherited by (0, 1). Now if A c (0, 1), let D(A) be the same as above but with only the points of A split; of course D(A) is just the quotient space of D obtained by identifying x~ with x+ for all x £ A. If A is uncountable, then D(A) is a compact perfectly normal nonmetrizable space.