We construct a compact Hausdorff space which has no P-points and yet in which every nonempty Gi set has nonempty interior. Definition 1 (Levy [1]). A space is an almost P-space if every nonempty (/?-set has nonempty interior. Definition 2. An element x of a topological space X is a P-point if x lies in the interior of each (/?-subset of X which contains it. An element x of a topological space X is a weak P-point if x lies in the closure of no countable subset of X {x}. In 1977 Levy [1] showed that every ordered compact almost P-space contains a P-point, and that every compact almost P-space of weight Ni contains a Ppoint. He asked whether there is any compact almost P-space which has no P-points. In 1978 Shelah [2] constructed a model of set theory in which co* has no P-points. Since Ε* is an almost P-space, this provided a consistent positive solution to Levy's question. We answer the question affirmatively in ZFC. Example 1 (ZFC). There is a compact almost P-space which contains no Ppoints. Let K β 2N' . Let P« be the family of all partial functions p whose domain is a countably infinite subset of zcU(zc x ojx) so that, if A 6 zc and a £ wx , then p(X) is either undefined or an element of 2 and p(X, a) is either undefined or an element of zc + 1 . Let tp be a mapping of k x cax onto Fn such that (k x 0)x ) n dom(tp(k, a)) C K x a. We use the notation (/, g) £ [p] to indicate that the total function (/, g) extends the partial function p . Let W be the set of (/, g) in 2K x(k+ l)**"" which satisfies the following Received by the editors July 30, 1992 and, in revised form, November 1, 1993. 1991 Mathematics Subject Classification. Primary 54D30, 54G10; Secondary 54D80. This work has been supported by the National Sciences and Engineering Research Council of Canada. © 1995 American Mathematical Society