This paper is a paraphrase and extension on my talk given at the conference \emph{Algebra, Logic and Dicrete Mathematics}, Niš, April 14--16, 1995, and it is inspired by Harte's theorem (Proc. Amer. Math. Soc. 99(1987), 328-330). In this paper we would like to present some results and problems connected with Harte's theorem.