In this paper we introduce the notion of an approximation space as a pair (U,

${\Pi})$, where U denotes an arbitrary, non-empty set and

${\Pi}$ denotes a covering of U. The elements of the covering

${\Pi}$ we call elementary sets in the approximation space (U,

${\Pi})$. We also introduce the notions of upper and lower approximation of a set in the space (U,

${\Pi})$, rough equality, rough inclusion, rough relation, and the notion of the approximation of a function. A theory of approximation which is based on these notions is a generalization of the theory of approximation in the sense of the papers by

*Z. Pawlak* [Int. J. Comput. Inf. Sci. 11, 341-356 (1982;

Zbl 0501.68053), Pr. Inst. Podstaw Inf. Pol. Akad. Nauk 435 (1981) and ibid. 467 (1981;

Zbl 0516.04001)] and the author [Demonstr. Math. 15, 1129-1133 (1982;

Zbl 0526.04005)]. In a special case when the covering

${\Pi}$ is a partition, these two theories are identical.