Type: Article

Topological propiertes of incomparable families

Campero-Arena G; Cancino, J; Hrusak, M...(et.al);

Abstract:

We study topological properties of families of mutually incomparable subsets of omega. We say that two subsets a and b of omega are incomparable if both a\b and b\a are infinite. We raise the question whether there may be an analytic maximal incomparable family and show that (1) it cannot be K-sigma, and (2) every incomparable family with the Baire property is meager. On the other hand, we show that non-meager incomparable families exist in ZFC, while the existence of a non-null incomparable family is consistent. Finally, we show that there are maximal incomparable families which are both meager and null assuming either r = c or the existence of a completely separable MAD family; in particular they exist if c < aleph(omega). Assuming CH, we can even construct a maximal incomparable family which is concentrated on a countable set, and hence of strong measure zero.