Ukrainian Mathematical Journal

, Volume 65, Issue 5, pp 799–805 | Cite as

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.


Continuous Function Topological Space Open Covering Hausdorff Space Product Topology 
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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • O. O. Karlova
    • 1
  • V. V. Mykhailyuk
    • 1
  1. 1.Fed’kovych Chernivtsi National UniversityChernivtsiUkraine

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