In this paper, we show that for an almost locally separable metrizable space X, if its Wijsman hyperspace induced by one compatible metric is Baire, then X itself must be Baire. We also provide an example to demonstrate the necessity of separability.

It is shown that for each k > 1, if f is a Baire one function and f is the product of k bounded Darboux (quasi–continuous) functions, then f is the product of k bounded Darboux (quasi–continuous) Baire one functions as well.

We define and investigate some new ideals of subsets of the Cantor space and the Baire space. We show that combinatorial properties of these ideals can be described by the splitting and reaping cardinal numbers. We show that there exist perfect Luzin sets for these ideals on the Baire space.

Let κ be a regular cardinal. Consider the Baire numbers of the spaces (2)κ for various θ ≥ κ. Let l be the number of such different Baire numbers. Models of set theory with l = 1 or l = 2 are known and it is also known that l is finite. We show here that if κ > ω, then l could be any given finite number. The Baire number of a topological space with no isolated points is the minimal cardinality ...

In this paper, we give alternative proofs of some of the properties of Baire-1 functions with respect to the new characterization of Baire-1 functions due to P.Y. Lee, W.K. Tang and D. Zhao. Some well-known functions were given to illustrate some of these properties. Mathematics Subject Classification: Primary 26A21

Introduction. Doob [3, p. 29],2 has shown that if ()= [U, 11, u] is a probability space (i.e. 11 is a cr-field of subsets of a set U and u is a countably additive non-negative measure defined on 11, normalized by the condition u(U) = l) and 1l0 is a o--subfield of 11, and y is an re-dimensional random variable mapping U into an re-dimensional Euclidean space X, then y possesses a conditional di...