Lebesgue measurability of separately continuous functions and separability

Abstract: It is studied a connection between the separability and the countable chain condition of spaces with the

L

-property (a topological space

X

has the

L

-property if for every topological space

Y

, separately continuous function

f : X i m e s Y o m a t h b b R

and open set

I s u b s e t e q m a t h b b R

the set

f^{- 1} (I)

is a

F_{s i g m a}

-set). We show that every completely regular Baire space with the

L

-property and the countable chain condition is separable and construct 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.

Cites Work