On symmetrically quasi-continuity and symmetrically cliquishness
by
Vasyl' Nesterenko
Chernivtsi National University

The notion of quasi-continuity was introduced by Kempisty [1] in 1932. Developing the results of Hahn and Banach he established that a separately quasi-continuous function is joint quasi-continuous. Later in 1953 Thielman [2] introduced the weaker property of quasi-continuity called cliquishness. It is easy to obtain that separately cliquish function doesn't need to be jointly cliquish.

Let (X, T_X) and (Y, T_Y) be topological spaces. A mapping f:X→ Y ((Y, r) is a metric space) is said to be quasi-continuous (cliquish) at a point x₀ ∈ X if for every set U ∈ T_X containing x₀ and for every set V ∈ T_Y containing y₀=f(x₀) (for each positive real e) there exists a nonempty set U₁ ∈ T_X such that U₁ ⊆ U and f(U₁) ⊆ V (r(f(x), f(x₀)) < e for every x ∈ U₁). If f is quasi-continuous (cliquish) at each point of its domain, it will be called quasi-continuous (cliquish).

Let (X, T_X), (Y, T_Y) and (Z, T_Z) be topological spaces. A mapping f:X×Y→ Z ((Z, r) is a metric space) is said to be symmetrically quasi-continuous (symmetrically cliquish) with respect to x at a point p₀=(x₀, y₀) ∈ X×Y if for arbitrary sets U×V ∈ T_X×T_Y containing p₀ and W ∈ T_Z containing z₀=f(p₀) (for each positive real e) there exist a set U₁ ∈ T_X containing x₀ and a nonempty set V₁ ∈ T_Y such that U₁×V₁ ⊆ U×V and f(U₁×V₁) ⊆ W (r(f(p), f(p₀)) < e for every p ∈ U₁×V₁). If f is symmetrically quasi-continuous (symmetrically cliquish) at each point of its domain then it is called symmetrically quasi-continuous (symmetrically cliquish).

Some results about the set of the points of symmetrically quasi-continuity of the jointly quasi-continuous mappings are obtained in [3]. Later this result was improved in [4], that is: if X is a topological space, Y and Z are second countable spaces and f:X×Y → Z is a jointly quasi-continuous mapping, then there exists a residual set A such that f is symmetrically quasi-continuous with respect to x at each point of A×Y. This result improves the theorem from [5] about the existence of the vertical lines of jointly quasi-continuous function f:R²→ R such that f is continuous with respect to the second variable.

We prove similar theorems for the cliquishness and results about the points of the symmetrically quasi-continuity of functions which are cliquish with respect to the one variable and quasi-continuous to the other variable.

Theorem 1. Let X be topological space, Y be a second countable space, Z be a metrizable space and f:X×Y→ Z be joint cliquishness. Then there is a residual set A such that f is symmetrically cliquish with respect to x at each point of A×Y.

We denote f^x(y)=f_y(x)=f(x, y).

Theorem 1. Let X be a topological space, Y be a second countable space, Z be a metrizable space, assume f:X×Y→ Z, f_y is cliquish for every y ∈ Y and f^x is quasi-continuous for every x ∈ X. Then there exists a residual set A such that f is symmetrically quasi-continuous with respect to x at each point of A×Y.

1. Kempisty S. Sur les fuctions quasicontinues, Fund. Math. - 1932. - 19. - P. 184-197.

2. Thielman H.P. Types of functions, Amer. Math. Monthly. - 1953. - 60. - P. 156-161.

3. Maslyuchenko V.K., Mykhajlyuk V.V., Nesterenko V.V. Symmetrical quasicontinuity of joint quasicontinuous functions, Mat. Stud. - 1999. - 9, ¹ 2. - P. 205-210.

4. Nesterenko V.V. On one characterization of joint quasi-continuity, Nauk. visn. Cherniv. un-tu. - Vyp. 336 - 337. Matematyka. - Chernivtsi: Ruta. - 2007. - P. 137 - 141 (in Ukrainian).

5. Kotlicka E., Maliszewski A. On quasi-continuity and cliquishness of sections of functions of two variables, Tatra Mt. Math. Publ. - 2002. - 24. - P. 105-108.

Date received: June 21, 2008