Topological structures, I

Abstract

The concept of a topological space has been a prime object of topological investigations. Unfortunately it suffers from certain deficiencies such as : (a) The category Top of topological spaces and continuous maps is not as well behaved as one would like it to be; e.g., Top is not cartesian closed, i.e., it is not possible to supply for any pair {X9 Y) of topological spaces the set X Y of all continuous maps from Y to X with a topology such that {X) is naturally isomorphic to X*. Also, in Top the product of quotient maps in general is no longer a quotient map. (b) Several important concepts of a topological nature—such as uniform convergence, uniform continuity, and completeness—cannot be expressed in the framework of the theory of topological spaces. There have been serious efforts by prominent mathematicians to remedy this situation. But none of the solutions offered is free from all the deficiencies mentioned above. The purpose of this note is to stimulate discussion on these matters among point set topologists.