Functions characterized by images of sets
by
Krzysztof Ciesielski
West Virginia University
Coauthors: D. Dikranjan, S. Watson

For non-empty topological spaces X and Y and arbitrary families \A subset or equal P(X) and B subset or equal P(Y) we put C_{A, B} = { f in Y^X : ( for allA in A) (f[A] in B) }. In the paper we examine which classes of functions F subset or equal Y^X can be represented as C_{A, B}. We are mainly interested in the case when F = C(X, Y) is the class of all continuous functions from X into Y. We prove that for non-discrete Tychonoff space X the class F = C(X, R) is not equal to C_{A, B} for any A subset or equal P(X) and B subset or equal P(R). Thus C(X, R) cannot be characterized by images of sets. We also show that none of the following classes of real functions can be represented as C_{A, B}: upper (lower) semicontinuous functions, derivatives, approximately continuous functions, Baire class 1 functions, Borel functions, and measurable functions.

Date received: February 9, 1997

Copyright © 1997 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 # caak-10.