Some new results on characterization of continuity
by
Giorgio Nordo
Universitá di Messina, Italy

The problem to characterize the continuity by images of sets was studied first in [V] by Velleman that have shown that the set C(R, R) of all real continuous functions on R can not be characterized by images of sets.

In [AP] Arenas and Puertas have demonstrated that - under some additional hypothesis - two classes of sets are sufficient to characterize continuity between two topological spaces. In fact, they have proved the following:

Theorem. Let X be a locally connected first countable space and let Y be a regular normal space. Then

C(X, Y) = CA, A \cap CB, B

where A is the family of all connected sets, B is the family of compact sets and C_{A, B} = { f in Y^X : f(A) in B for every A in A }.

Here we will present two generalizations of this result for:

• the class of q-spaces (introduced by Michael in [M]) and
• the class of sequential spaces.

References

[AP] F.G. Arenas, M.L. Puertas, Characterizing continuity in topological spaces, preprint (1998).

[M] Michael E., A note on closed maps and compact sets, Israel J. Math., 2, 3 (1964), 173-176.

[V] Velleman D.J., Characterizing continuity, American Math. Monthly, 104, 4 (1997), 318-322.

Date received: July 8, 2000