English

The discontinuity points set of separately continuous functions on the products of compacts

General Topology 2015-12-25 v1

Abstract

It is solved a problem of construction of separately continuous functions on the product of compacts with a given discontinuity points set. We obtaine the following results. 1. For arbitrary \v{C}ech complete spaces XX, YY and a separable compact perfect projectively nowhere dense zero set EX×YE\subseteq X\times Y there exists a separately continuous function f:X×YRf:X\times Y\to\mathbb R the discontinuity points set of which equals to EE. 2. For arbitrary \v{C}ech complete spaces XX, YY and nowhere dense zero sets AXA\subseteq X and BYB\subseteq Y there exists a separately continuous function f:X×YRf:X\times Y\to\mathbb R such that the projections of the discontinuity points set of ff coincide with AA and BB respectively. An example of Eberlein compacts XX, YY and nowhere dense zero sets AXA\subseteq X and BYB\subseteq Y such that the discontinuity points set of every separately continuous function f:X×YRf:X\times Y\to\mathbb R does not coincide with A×BA\times B, and CHCH-example of separable Valdivia compacts XX, YY and separable nowhere dense zero sets AXA\subseteq X and BYB\subseteq Y such that the discontinuity points set of every separately continuous function f:X×YRf:X\times Y\to\mathbb R does not coincide with A×BA\times B are constructed.

Keywords

Cite

@article{arxiv.1512.07758,
  title  = {The discontinuity points set of separately continuous functions on the products of compacts},
  author = {V. V Mykhaylyuk},
  journal= {arXiv preprint arXiv:1512.07758},
  year   = {2015}
}