Topological Constructs

In order to handle problems of a topological nature topologists have created not only topological spaces but also uniform spaces, filter spaces, convergence spaces and so on. Since constructions in the corresponding concrete categories of these spaces have striking similarities the question arises whether it is possible to postulate axioms for a construct (= concrete category) which may be regarded as topological. Thus, the problem consists in looking for one or more properties which are independent of the special structure of the considered objects in a construct (i.e. properties essentially characterized by morphisms) and which are not satisfied by “algebraic” constructs. This claim is fulfilled by the initial structures in the sense of N. Bourbaki [18] provided their unrestricted existence is required. In the category Group for instance there do not exist arbitrary initial structures, e.g. not every subset of a group is a subgroup. Further conditions may be added for getting the concept “topological construct” but they are of a more “technical” nature. In order to obtain final structures simultaneously it is useful (in constrast to N. Bourbaki) to require the existence of initial structures for families of maps which are indexed by a class (instead of a set). After the definition of a topological construct and numerous examples (up to measure theory and algebraic topology) the categorical properties of topological constructs are studied in this chapter.