Characterization of function classes C(Y)|X
by
E. Makai, Jr.
Mathematical Institute of the Hungarian Academy of Sciences
Coauthors: Á. Császár

Let X be a topological space and let F Ì C(X). Then there exists a topological space Y containing X as a subspace and such that F = C(Y)|X, if and only if F is weakly composition closed, i.e., for any index set I, any f_i Î F (i Î I) and any continuous map k: R^I ® R we have k °áf_i ñ Î F, where áf_iñ:X ® R^I is the map with i-th coordinate f_i. The analogous statement is valid for functions to any T₁ space, rather than to R, and even we can consider functions to any set of T₁ spaces, and then a generalization of the above statement is valid, with a suitably defined weak composition closedness property. We also show that some earlier results on characterization of function classes F Ì C(X) of the form C(Y)|X, with Y some extension of a given topological space X, and on the characterization of function classes C(áX, T ñ), with T some topology on a given set X, respectively, can be generalized in an analogous way as above, by means of composition properties analogous to the above one or by filter closedness (for functions to any set of T₃ spaces, or to any set of topological spaces, respectively).

[1]Á. Császár, E. Makai, Jr., Characterization of function classes C(Y)|X, Acta Math. Hungar. 81, 1998.

Date received: March 4, 1998