Continuity between spaces defined on Seq
by
Jerry E. Vaughan
University of North Carolina at Greensboro

Let Seq denoted the set of all finite sequences of natural numbers (Seq = \cup _{n in \omega}ⁿ\omega). In [Advances in Mathematics, 29 (1978) 89 - 130], V. Kannan and M. Rajagopalan considered a certain countable space that is now often denoted by Seq. Their topology on Seq is defined using a family, {u_t: t in Seq}, of ultrafilters on the set \omega of natural numbers. We use the definition of the topology on Seq as given (independently) by A. Szymanski: A set U subset Seq is open if and only if for every t in U, {n : t \cup {(dom(t), n)} in U} in u_t. Let <= _RK denote the Rudin-Keisler order on ultrafilters. Two ultrafilters u, v are said to have the same type provided u <= _RK v and v <= _RK u; otherwise u, v are said to have different type.

The following theorem and example improve a result of Kannan-Rajagopalan.

Theorem If f:Seq(u_t) --> Seq(v_t) is continuous and v_f(t) \not <= _RK u_t for all t in Seq, then f is locally constant on Seq.

Example There exists a set of ultrafilters {u_t, v_t : t in Seq} having pairwise different types and a function f:Seq(u_t) --> Seq(v_t) such that f is continuous (and open) and not locally constant at any t in Seq.

The example shows that the theorem cannot be improved by replacing the condition `` v<sub>f(t)</sub> \not <= <sub>RK</sub> u<sub>t</sub>'' by the weaker condition ``u_t and v_f(t) have different type.''

Date received: February 13, 2000