# 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.

Published

25.05.2013

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 65, no. 5, May 2013, pp. 722–727, http://umj.imath.kiev.ua/index.php/umj/article/view/2456.

Issue

Section

Short communications