On Linear Continuous Open Surjections of the Spaces C_p(X)
by
Arkady Leiderman
Ben-Gurion University
Coauthors: M. Levin

We give the negative answer to the following problem of Arkhangel'skii (Open Pronblems in Topology, Problem 1047): assume that C_p(X) can be mapped by a linear continuous open mapping onto C_p(Y) for compact spaces X and Y. Is it true that dimension of Y is less or equal than dimension of X?

Theorem 1 For every finite dimensional compactum Y there exists a 2-dimensional compactum X which admits a linear continuous open surjection from C_p(X) onto C_p(Y).

We conjecture that X in Theorem 1 can be replaced by the unit closed interval [0, 1] but we do not know even if dim X can be reduced to 1. In this direction we prove the following

Theorem 2 For every natural n there exist n-dimensional compactum Y and 1-dimensional compactum X such that C_p(X) admits a canonical linear continuous open surjection onto C_p(Y).

In general, linear continuous surjections of C_p-spaces fail to be open.

Example 1. There exists a linear continuous surjection of C_p[0, 1] onto itself which is not open.

Date received: August 14, 1996

Copyright © 1996 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 Mathematical Conference Abstracts. Document # caaj-71.