On iterated dualizations of topological spaces and structures
by
Martin Maria Kovar
Technical University of Brno, Czech Republic

Recall that a topology is said to be dual with respect to the original topology on a set if it has a closed base consisting of compact saturated sets. In the well-known book Open Problems in Topology, edited by J. van Mill and G. M. Reed, there was stated (among many other, no less interesting problems) a problem no. 540 of J. D. Lawson and M. Mislove: Does the process of iterating duals of a topology terminate by two topologies, dual to each other (1990)?

It should be noted that this problem was solved, for T1 spaces, a long time before it was formulated by Lawson and Mislove, by G. E. Strecker, J. de Groot and E. Wattel (1966). Namely, in T1 spaces, the dual operator studied by Lawson and Mislove coincide with another dual, introduced by de Groot, Strecker and Wattel more than 30 years ago. In 2000 the problem was partially solved by B. Burdick, who proved that for some topologies on certain hyperspaces, during the iterated dualization process there can arise at most four distinct topologies: the original topology, then the first dual, the second dual and the third dual, since the fourth dual topology coincides with the second one. Finally, this result was generalized for all topological spaces by M. M. Kovar (2001). In this talk we will discuss the following rings of questions:

(1) We will present some recent results related to iterated dualizations of topological spaces

(2) We will ask what happens with the dualizations if we leave the realm of spatiality.

(3) We will mention some (unsolved) problems related to dual topologies.

Date received: April 11, 2002