On two classes of functions with the Hahn property
by
Vasyl Nesterenko
Chernivtsi National University

A function f:X×Y → Z is defined to have the Hahn property if there is a residual subset A of X such that A×Y ⊆ C(f).

In [1] J.Calbrix and J.Troallic proved that a separately continuous functon f:X×Y→ Z has the Hahn property if X is a topological space, Y is a second countable space and Z is a metrizable space. In [2] and [3] this result was generalized to other classes of functions, in particular, to the class K_hC of functions which are horizontally quasicontinuous with respect to the first variable and continuous with respect to the second variable.

Those results suggested to introduce a new class of spaces. A topological space Y is defined to be a Hahn space if for any topological space X and any metrizable space Z a function f:X×Y→ Z has the Hahn property provided f is continuous with respect to the second variable and continuous with respect to the first variable for values of the second variable that belong to a dense subset of Y. Each second countable space is Hahn, but the converse is not true.

Theorem 1 Let X be a topological space, Y is a Hahn first countable separable space, Z is a metrizable space. A function f:X×Y → Z has the Hahn property provided f is jointly quasicontinuous and almost continuous with rspect to the second variable for values of the first variable that belong to some residual set.

Professor O.Skaskiv posed a problem of description of the structure of the set of discontinuity points of separately monotone function f:R²→R. It can be proved that such functions are poinwise discontinuous. However the complete description of the set of discontinuity points of such function is still not known. We were able to prove the following theorem.

Theorem 2 Let Y be a separable first countable Hahn space. A function f:R×Y → R has the Hahn property provided f is monotone with respect to the first variable and continuous with respect to the second variable for values of the first variable that belong to some residual subset of X.

References:

[1] Calbrix J., Troallic J.-P. Applications separement continues // C.R.Acad. Sc.Paris. Sér.A. 288 (1979), 647 - 648.

[2] Maslyuchenko V.K. Hahn spaces and a Dini problem // Mat. Metody i fix.-mekh. polia, 41:4 (1998), 39- 45 (in Ukrainian).

[3] Maslyuchenko V.K., Nesterenko V.V. Joint continuity and quasicontinuity of horizontally quasicontinuous functions // Ukr. Mat. Zhurn. 52:12 (2000), 1711-1714 (in Ukraininan).

Date received: May 6, 2008