Exponents of someN-compact spaces

Abstract

For any topological spaceT, S. Mrówka has defined Exp (T) to be the smallest cardinal κ (if any such cardinals exist) such thatT can be embedded as a closed subset of the productN ^κ of κ copies ofN (the discrete space of cardinality ℵ₀). We prove that forQ, the space of the rationals with the inherited topology, Exp (Q) is equal to a certain covering number, and we show that by modifying some earlier work of ours it can be seen that it is consistent with the usual axioms of set theory including the choice that this number equal any uncountable regular cardinal less than or equal to 2^ℵ ₀. Mrówka has also defined and studied the class ℳ={κ: Exp (N _κ)=κ} whereN _κ is the discrete space of cardinality κ. It is known that the first cardinal not in ℳ must not only be inaccessible but cannot even belong to any of the first ω Mahlo classes. However, it is not known whether every cardinal below 2^ℵ ₀ is contained in ℳ. We prove that if there exists a maximal family of almost-disjoint subsets ofN of cardinality κ, then κ∈ℳ, and we then use earlier work to prove that if it is consistent that there exist cardinals which are not in the first ω Mahlo classes, then it is consistent that there exist such cardinals below 2^ℵ ₀ and that ℳ nevertheless contain all cardinals no greater than 2^ℵ ₀. Finally, we consider the relationship between ℳ and certain “large cardinals”, and we prove, for example, that if μ is any normal measure on a measurable cardinal, then μ(ℳ)=0.