The role of Nearness in convenient topology
by
Gerhard Preuß
FU Berlin

Convenient Topology consists essentially in the study of semiuniform convergence spaces which form a cartesian closed and hereditary topological construct with the additional property that products of quotients are quotients, i.e. a strong topological universe, denoted by SUConv. Furthermore, convergence structures and uniform convergence structures (including topological and uniform structures) are available, and initial and final structures have an easy description! Via the subconstruct Fil of filter spaces, which form the link between convergence structures and uniform convergence structures, SUConv is related to the construct Mer of merotopic spaces and via the subconstruct SubTop of subtopological spaces to the construct Near of nearness spaces, namely in both cases by means of bicoreflective embedding. Since topological spaces behave badly with respect to the formation of subspaces, much better results are obtained by forming subspaces of (symmetric) topological spaces in SUConv, because e.g. in the framework of semiuniform convergence spaces subspaces of normal (resp. paracompact) spaces are normal (resp. paracompact). Even dimension theory (including cohomological dimension theory) profits from this better behavior of subspaces. But the decisive step for obtaining these results is the above mentioned relation to nearness spaces. Last but not least, the question how the subspaces, formed in Fil (resp. SUConv) of compact symmetric topological spaces regarded as filter spaces (resp. semiuniform convergence spaces), called subcompact spaces, can be characterized axiomatically and the corresponding question for compact Hausdorff spaces are solved. For the proof of the characterization of subcompact spaces the Herrlich completion of nearness spaces is used, whereas for the proof of the corresponding characterization of the so-called sub-(compact Hausdorff) spaces the Hausdorff completion of uniform spaces suffices, since the construct of sub-(compact Hausdorff) spaces is concretely isomorphic to the construct SepProx of separated proximity spaces (proximity spaces are identified with totally bounded uniform spaces).

Date received: June 16, 2000

Copyright © 2000 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 # caeq-47.