Sequential closure of the space of jointly continuous functions in the space of separately continuous functions

  • H. A. Voloshyn
  • V. K. Maslyuchenko

Abstract

Given compact spaces $X$ and $Y$, we study the space $S(X \times Y )$ of separately continuous functions $f : X \times Y \rightarrow R$ endowed with the locally convex topology generated by the seminorms $|| f||^x = \mathrm{max}_{y \in Y} |f(x, y)|,\; x \in X$, and $|| f||_y = \mathrm{max}_{x \in X} |f(x, y)|,\; y \in Y$. Under the assumption that the compact space $X$ is metrizable, we prove that a separately continuous function $f : X \times Y \rightarrow R$ is the limit of a sequence $(f_n)^{\infty}_{n=1}$ of jointly continuous function $f_n : X \times Y \rightarrow R$ in $S(X \times Y )$ provided that the set $D(f)$ of discontinuity points of $f$ has countable projections on $X$.
Published
25.02.2016
How to Cite
VoloshynH. A., and MaslyuchenkoV. K. “Sequential Closure of the Space of Jointly continuous functions in the Space of Separately Continuous Functions”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 68, no. 2, Feb. 2016, pp. 156-61, http://umj.imath.kiev.ua/index.php/umj/article/view/1830.
Section
Research articles