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Organizers |
Successive topologizations and regularizations
by
Szymon Dolecki
Mathematical Institute of Burgundy, Burgundy University, Dijon, France
Coauthors: H.-P. Künzi (Rondebosch), T. Nogura (Matsuyama)
L. Foged proved that a weakly regular topology on a countable set is regular. In terms of convergence theory, this means that the topological reflection Tx of a regular pretopology x on a countable set is regular. It is proved that this still holds if x is a regular s-compact pretopology. On the other hand, it is proved that for each finite n there is a (regular) pretopology r (on a set of cardinality continuum) such that (RT)kr > (RT)nr for each k < n and (RT)nr is a Hausdorff compact topology, where R is the reflector to regular pretopologies. The proof is based on a concatenation of modules constructed with the aid of a MAD family of Pétr Simon. It is also shown that there exists a regular pretopology of Hausdorff RT-order greater than the first countable ordinal. Moreover, all these pretopologies have the property that all the points except one are topological and regular.
Date received: January 21, 2009
Copyright © 2009 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caxy-20.