Volterra versus Baire spaces?

Abstract

A topological space X is called Volterra if for each pair of real-valued functions f,g: X → R such that C(f) and C(g) are dense, C(f)∩C(g) is also dense, where C(f) and C(g) are sets of points of continuity of f and g respectively. It is clear that every Baire space is Volterra. There are many Volterra spaces which are not Baire, for example, a first countable, completely regular, paracompact space that is Volterra and is not Baire was construced by Gruenhage and Lutzer in 2000. In this talk, I shall discuss when a Volterra space is Baire. Our first result claims that every stratifiable Volterra space is Baire. This answers affirmatively an open question posed by Gruenhage and Lutzer in 2000. Further, we establish that a locally convex topological vector space is Volterra if and only if it is Baire; and the weak topology of a topological vector space fails to be Baire if the dual of the space contains an infinite linearly independent pointwise bounded subset. Consequently, a normed linear in its wesk topology is Volterra if and only if it is finite-dimesnional. This extends a classical result, which says a normed linear in its wesk topology is Baire if and only if it is finite-dimesnional.