Cross Topology and Lebesgue Triples

  • O. O. Karlova
  • V. V. Mykhailyuk


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.
How to Cite
KarlovaO. O., and MykhailyukV. V. “Cross Topology and Lebesgue Triples”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, no. 5, May 2013, pp. 722–727,
Short communications