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

Abstract

We investigate the existence of a separately continuous function f : X × Y → R with a onepoint set of discontinuity points in the case where the topological spaces X and Y satisfy conditions of compactness type. In particular, it is shown that, for compact spaces X and Y and nonisolated points x 0 ∈ X and y 0 ∈ Y, a separately continuous function f : X × Y → R with the set of discontinuity points { ( x 0 , y 0 ) } exists if and only if there exist sequences of nonempty functionally open sets in X and Y that converge to x 0 and y 0 , respectively. 1.It follows from the Namioka theorem [1] that the set of discontinuity points of a separately continuous function, i.e., a function continuous in each variable, in particular on the product of two compact spaces, is contained in the product of sets of the first category. 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. This result was generalized in [4].Conditions on topological spaces X and Y and points a ∈ X and b ∈ Y that guarantee the existence of a separately continuous function f : X × Y → R with the set of discontinuity points { ( a, b ) } were studied in [5] (see also [6]). It was shown there that, for the purpose indicated, it is sufficient that X and Y be of the Tikhonov type and a and b be nonisolated G δ -points in the corresponding spaces; in this case, the space Y should either have a countable base of neighborhoods of the point b or be locally connected at the point b. Moreover, it follows from the results of [7], where the problem of the construction of a separately continuous function on the product of Eberlein compact spaces with a given set of discontinuity points was solved, that the statement formulated above is also true in the case where X and Y are Eberlein compact spaces and a ∈ X and b ∈ Y are nonisolated points. 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.The present paper is devoted to the investigation of this problem. ...

Metrizable compacta in the space of continuous functions with the topology of pointwise convergence

“…Let X be a set and g : X × Y → Z. Then g is concentrated on a set S ⊆ T in the second variable if g(x, y ′ ) = g(x, y ′′ ) for any x ∈ X and y ′ , y ′′ ∈ Y with y ′ | S = y ′′ | S ; and g depends upon κ coordinates in the second variable if |S| ≤ κ for some S. The following result was obtained in [9] for κ = ℵ 0 . Proof.…”

The discontinuity points set of separately continuous functions on the products of compacts

“…The problem of construction of separately continuous function on the product of two compact spaces with a given one-point discontinuity points set was solved in [7] using a dependence of functions upon some quantity of coordinates technique. It was obtained in [7] that for nonisolated points x 0 and y 0 in compact spaces X and Y respectively there exists a separately continuous function f :…”