Walter P. Tholen - Topology based on maps

Compared with the large variety of notions on spaces, maps seem to play a minor role in General Topology. We wish to show that it is beneficial to think of fundamental object notions as being induced by particular classes of morphisms. Hence, with any class of morphisms in a category satisfying some weak but essential properties we associate two new classes of morphisms, and , and two classes of objects, and , such that, under suitable restriction, both C and D belong to known and fundamental Galois correspondences. For example, for the class of closed continuous maps in the category of topological spaces, and are the classes of proper and of separated maps, respectively, and and are the categories of compact spaces and of Hausdorff spaces, respectively. Other interesting choices for would include the classes of open maps and of dense maps. Among other things we wish to

  explore the interaction of the classes and and derive their standard properties which, in the paradigmatic example, gives a surprisingly complete theory of compactness/Hausdorff separation/perfectness

  discuss with the same general methods connectedness/total disconnectness, both at the object and map levels

  show that the required axioms are satisfied for the three basic choices for (as mentioned above for spaces) and are actually available in any category which comes equipped with a (reasonably good) closure operator

  demonstrate that the morphism-object interaction is advantageously reversed by slicing the categories in question, giving us also a safe guideline for the ``right'' fibrewise notions

  apply the theory in particular to Diers' Zariski closure for algebraic sets (with respect to any monad over sets).