Stationary sets and on the existence of homeomorphisms between them

Abstract

Stationary sets are important tools in proofs of properties in sets of uncountable cardinality. In this thesis we look at mapping properties between stationary sets. First, the theory necessary for the construction and evaluation of stationary sets is made. That is the theory of ordinal, cardinal and regular cardinal numbers is build up from the level of knowledge of a mathematics student. Two important theorems for stationary sets, Fodor's theorem and the theorem of Ulam and Solovay, are proven. Next mapping properties of stationary subsets of a regular cardinal $\kappa$ under measurable functions is looked into. With these properties we construct a necessary condition for the existence of homeomorphisms between stationary sets; they may only differ on a non-stationary set. Lastly the amount of stationary subsets of a regular cardinal k without a homeomorphism between them is estimated as the cardinality of the power set of k. We find that there are 2 to the power k topologically incomparable subsets of k.