Recent Progress in General Topology

Introduction The Cech-Stone compactiication of the real line R seems to have been much less investigated than the space !. One reason for this is that ! can be attacked with two kinds of weapons: Boolean algebraic and topological. The space R is a continuum, so it is not susceptible to Boolean algebraic treatment. Be that as it may, R deserves more of our attention than it has received so far. In an eeort to catch this attention I have tried to give a coherent overview of our knowledge of R at this time. There have been several interesting developments in R recently. As an answer to a well-known question of van Douwen, Yu 1991] constructed an autohomeo-morphism of R that maps a remote point to a non-remote point. Another recent result is actually about ! ; it is well-known that the principle NCF implies that ! can be covered by nowhere dense P-sets; in 1991a] Zhu has shown that NCF in fact implies that ! can be covered by a chain of nowhere dense P-sets. The proof makes essential use of the structure of the family of subcontinua of R. Acknowledgements. I would like to thank several people for providing me with material to write about. Jan van Mill kindly gave me access to the unpublished work of Eric van Douwen (hereafter cited as`van Douwen's notes')