Compact topological spaces inspired by combinatorial constructions
Due to Mr�owka [24], polyadic spaces are compact Hausdor� spaces that are continuous images of some power of the one point compactication� of a discrete space �. It turns out that many results about polyadic spaces hold for a more general class spaces, as we shall show in this thesis. For a sequence� = h�i : i 2 Ii of cardinals, a compact Hausdor� space X is �-multiadic if it is a continuous image of Yi2I��i. It is easy to observe that a �-multiadic space is �-polyadic, but whether the converse is true is a motivation of this dissertation. To distinguish the polyadic spaces and multiadic spaces, we consider (��)I and Yi2I ��i. We investigate two cases regarding �: if it is a successor or a limit cardinal. For an inaccessible cardinal � we clarify by an example that the polyadic space (��)� is not an image of Yi<���i. Beside this result we and a model of set theory using Prikry-like forcing to get an analogous result when � is singular. Although the individual polyadic and multiadic spaces differ, we show that the class of polyadic spaces is the same as multiadicclass! Moreover, this dissertation is concerned with the combinatorics of multiadic class that can be used to give some of their topological structure. We give a Ramsey-like property for the class of multiadic compacta called Q� where � is a regular cardinal. For Boolean spaces this property is equivalent to the following: every uncountable collection of clopen sets contains anuncountable subcollection which is either linked or disjoint. We give gen- eralizations of the Standard Sierpi�nski graph and use them to show that the property of being �-multiadic is not inherited by regular closed sets for arbitrarily large 2
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