Tutorial in Set-Theoretic Topology
by
Peter Nyikos
University of South Carolina

This tutorial is designed to give a feel for the techniques and topics in set-theoretic topology through the presentation of some standard tools and examples. The axiom of choice (in the forms of Zorn's Lemma and well-ordering) will be used to construct some examples, while others will require the use of axioms independent of the usual ones, such as the continuum hypothesis (CH) and the combinatorial principles ♣ and ♢. These constructions in turn will be contrasted with theorems using such tools as Martin's Axiom (MA) together with the negation of CH.

Most of the topological properties involved will be quite easy to define. In fact, of the charms of set-theoretic topology is that it settles a great many relationships between fundamental topological concepts. For instance, take the following simple question: is every perfectly normal, countably compact space compact? [A perfectly normal space is a normal-this includes Hausdorff-space in which every closed set is a countable intersection of open sets.] The axiom ♢ can be used to construct a counterexample. On the other hand, MA + not-CH implies that every perfectly normal, countably compact space is compact. It is also noteworthy that even CH is insufficient to settle this question either way: on the one hand, ♢ implies CH; on the other hand, there is a model of CH in which every perfectly normal, countably compact space is compact.

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Date received: June 15, 2008