International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 54159, 4 pages
doi:10.1155/2007/54159

Lebesgue measurability of separately continuous functions and separability

Abstract

A connection between the separability and the countable chain condition of spaces with L-property (a topological space X has L-property if for every topological space Y, separately continuous function f:X×Y→ℝ and open set I⊆ℝ, the set f−1(I) is an Fσ-set) is studied. We show that every completely regular Baire space with the L-property and the countable chain condition is separable and constructs a nonseparable completely regular space with the L-property and the countable chain condition. This gives a negative answer to a question of M. Burke.