2018
Том 70
№ 8

# Cross Topology and Lebesgue Triples

Abstract

The cross topology γ on the product of topological spaces X and Y is the collection of all sets G ⊆ X × Y such that the intersections of G with every vertical line and every horizontal line are open subsets of the vertical and horizontal lines, respectively. For the spaces X and Y from a class of spaces containing all spaces ${{\mathbb{R}}^n}$ , it is shown that there exists a separately continuous function f : X × Y → (X × Y, γ) which is not a pointwise limit of a sequence of continuous functions. We also prove that each separately continuous function is a pointwise limit of a sequence of continuous functions if it is defined on the product of a strongly zero-dimensional metrizable space and a topological space and takes values in an arbitrary topological space.

English version (Springer): Ukrainian Mathematical Journal 65 (2013), no. 5, pp 799-805.

Citation Example: Karlova O. O., Mykhailyuk V. V. Cross Topology and Lebesgue Triples // Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 722–727.

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