Inverse problems of the theory of separately continuous mappings

Quasicontinuous and separately continuous functions with values in Maslyuchenko spaces

“…By definition of the cardinal add(M), on the real line there is a family A consisting of |A| = add(M) meager sets such that the union A is not meager in R. Replacing each A ∈ A by a larger meager set, we can assume that A is of type F σ in R. Also we can assume that the family A is closed under finite unions. By a classical result of Baire (see also [24]), for any meager Using Propositions 5.5 and 5.3, we shall prove that the class of Maslyuchenko spaces is closed under Σ <κproducts with κ ≤ add(M). For a family (X α , * α ), α ∈ A, of pointed topological spaces and a cardinal κ the subspace Σ <κ α∈A (X α , * α ) = {(x α ) α∈A ∈ α∈A X α : |{α ∈ A : x α = * α }| < κ} of the Tychonoff product α∈A X α will be called the Σ <κ -product of the family of pointed spaces X α , α ∈ A.…”

One-Point Discontinuities of Separately Continuous Functions on the Product of Two Compact Spaces

“…There naturally arises the problem of the description of the sets of discontinuity points of separately continuous functions on products of compact spaces (see [2]); in particular, the following question arises: Is it true that every one-point set with nonisolated projections in the product of two compact spaces is the set of discontinuity points of some separately continuous function? In [3] (Theorem 9), it was shown that the answer to this question is negative. Namely, on the product of two Tikhonov cubes at least one of which has an uncountable weight, a separately continuous function with one-point discontinuity does not exist.…”

“…Note that the main tool used in the proof of the results of [7] is a Preiss-Simon-type property of Eberlein compact spaces (see [8, p. 170]), i.e., the presence of a sequence of open sets that converges to a given point in an Eberlein compact space. Since a Tikhonov cube of uncountable weight does not possess this property, on the basis of Theorem 9 in [3] it is natural to establish whether the existence of convergent sequences of open sets in compact spaces X and Y is a necessary condition for the existence of a separately continuous function f : X × Y → R with one-point discontinuity set.…”

Characterization of the sets of discontinuity points of separately continuous functions of many variables on the products of metrizable spaces

“…Some new approaches to the solution of the inverse problem on the product of metrizable spaces were proposed in [4,5,6]. The method from [6] used locally finite systems and Stone's theorem on the paracompactness of a metrizable spaces.…”