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Ukrainian Mathematical Journal

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

Cross Topology and Lebesgue Triples

  • O. O. Karlova
  • V. V. Mykhailyuk
Article
  • 46 Downloads

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.

Keywords

Continuous Function Topological Space Open Covering Hausdorff Space Product Topology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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