Topologically Inseparable Functions
by
Renato Lewin
Pontificia U. Catolica de Chile

In Walter's " orange monograph" The Clone of a Topological Space, p. 24, the following question is posed: which equations can be modelled on (the topological space) A by continuous operations? A related question is: can one describe the clone of operations of a given (topological) algebra? These questions motivate the problem I will talk about.

Suppose f is your favourite selfmap of a set A, (for instance a unary operation of the algebra A.) Given a topology on A for which f is continuous, there will be other functions that are also continuous, for instance, constant functions and iterates of f, but in a sense, these are trivially continuous. A very natural question arises then.

Are there any non-trivial selfmaps g that are continuous for every topology for which f is continuous?

Intuitively, these maps g cannot be separated from f by a topology. We give conditions on the function f so that the only such g are the above mentioned trivial functions.

Date received: July 9, 2004

Copyright © 2004 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 # caoc-25.