• DOI:10.2307/2048389
• Corpus ID: 123600219

Large Compact Separable Spaces May all Contain βN

@inproceedings{Dow1990LargeCS,
  title={Large Compact Separable Spaces May all Contain $\beta$N},
  author={Alan Dow},
  year={1990}
}

• A. Dow
• Published 1 May 1990
• Mathematics

In the Cohen model any compact separable space that does not contain ßN has cardinality at most of the continuum. In this paper we prove the theorem stated in the abstract. By the Cohen model we mean any model obtained by adjoining N2 Cohen reals to a model of GCH. We shall, in fact, show that any compact separable space of cardinality greater than the continuum c will map onto the space /' . We will see that this is equivalent to containing ßN. We freely use any notation and conventions which we believe to be standard enough to justify this usage. Suppose that F is a model of GCH, G is Fn(co2, 2)-generic over V, and that in V[G], X is a compact separable space with cardinality greater than c. We will assume (and show later that we may assume) that X has a countable discrete dense set co. In V, let 6 be a large enough regular cardinal and let M be an elementary submodel of 7/(0) such that X g M[G], \M\ = N2 and such that Mw< c M. It follows that M[G]W] c M[G]. Still in V, choose an elementary chain {M(>\a < co2} of elementary submodels of M whose union is M and so that each submodel is closed under co-sequences and each has cardinality N, . It is remarked in [2] that since M < H = Hid), it will be the case that M [G] < H[G] in V[G]. This is easily proven using names, forcing, and the definition M[G] = {val (i, C7)|i G M is a Fnico-,, 2)-name}. It will also be the case here (although not in general) that H[G] will be //(Ö) as computed in V[G]. Passing to V[G], choose a point x G X\M . For each q < cd, , define x" = {Y e 9>ico) n Mn[G]\x e intv clA. F}. Although x is not in M[G], it is in H[G], and the set x" is just an w,-sized subset of Mn[G] and is therefore in M [G]. By elementarily, it follows that Received by the editors June 14, 1989 and, in revised form, August 7, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 54A25, 54A35.