Two remarks about analytic sets 1 Fons van Engelen 2

In this paper we give two results about analytic sets. The rst is a counterexample to a problem of Fremlin. We show that there exists! 1 compact subsets of a Borel set with the property that no-compact subset of the Borel set covers them. In the second section we prove that for any analytic subset A of the plane either A can be covered by countably many lines or A contains a perfect subset P which does not have three collinear points. In his book about Martin's Axiom [2] p. 61 D. H. Fremlin shows that, assuming MA, for any analytic set X and set A X of cardinality less than the continuum there exists a-compact set L such that A L X. (-compact means countable union of compact sets.) Here we show that the 1in Lecture Notes in Mathematics, Springer-Verlag, 1401(1989), 68-72. 2Research partially supported by the Netherlands organization for the advancement of pure research 3Research partially supported by NSF grant MCS-8401711 1 set A cannot be replaced by a family of compact sets of cardinality! 1. This answers a question of Fremlin [2] p.67. Let Q be the rationals and let E = Q! 0. E is a an F set. For each <! 1 let C be a compact subset of Q which is homeomorphic to + 1 and let H = C!