9312363 Arhangel'skii The main objective of this project is to understand how closely countably compact spaces or groups of countable tightness resemble compact spaces or groups. The project is organized around instances of the following two general questions: Which results on cardinal functions of compact spaces generalize to countably compact spaces, and under which conditions do the latter properties (at least consistently) already imply compactness of the space? Which results on compact spaces and continuous mappings can be generalized to countably compact spaces if the space is a topological group (and the mapping is also a homomorphism)? The concept of compactness of a topological space lies at the heart of General Topology; it is its most important idea from the point of view of applications. In order to understand the notion of compactness fully, one has also to study weakenings of the compactness property. Perhaps the most important such weakening is a property called countable compactness. The aim of the project is to determine which theorems about compact spaces and continuous functions between compact spaces remain valid if the spaces in question are only assumed to be countably compact instead of compact. A similar investigation will be done for topological groups, that is, for spaces which in addition to the topology also carry the structure of an algebraic group. ***
