One-Point Discontinuities of Separately Continuous Functions on the Product of Two Compact Spaces

  • V. V. Mykhailyuk

Abstract

We investigate the existence of a separately continuous function $f :\; X \times Y \rightarrow \mathbb{R}$ with a one-point set of points of discontinuity in the case where the topological spaces $X$ and $Y$ satisfy conditions of compactness type. In particular, for the compact spaces $X$ and $Y$ and the nonizolated points $x_0 \in X$ and $y_0 \in Y$, we show that the separately continuous function $f :\; X \times Y \rightarrow \mathbb{R}$ with the set of points of discontinuity $\{(x_0, y_0)\}$ exists if and only if sequences of nonempty functionally open set exist in $X$ and $Y$ and converge to $x_0$ and $y_0$, respectively.
Published
25.01.2005
How to Cite
Mykhailyuk, V. V. “One-Point Discontinuities of Separately Continuous Functions on the Product of Two Compact Spaces”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, no. 1, Jan. 2005, pp. 94–101, https://umj.imath.kiev.ua/index.php/umj/article/view/3576.
Section
Research articles