In the paper [3], the author suggested a general topological approach to domain theory as highly convenient and more general than the established more traditional approach using dcpos (directed-complete partial orders) starting from D. Scott’s work. This approach was realized by the author in the papers [2, 3] for the cases of f-spaces and A-spaces (complete f&space= algebraic bounded-complete domain, complete Aospace= bounded-complete domain; in the sequel, the term bc-domain will be used to denote bounded-complete domains). In the introduction to [3], the properties of the relation 4 of a recognizable approximation was discussed. I would like to quote from [3]:“It is natural to require that all recognizable approximations of a fixed element x should form a directed set and this can be satisfied in the strong (but sufficiently reasonable) form:(5) if x0+ x and x1 3 x then there exists an element x2 E X0 [basis] which is the exact upper bound (x2= x0 V XI) of these elements in (X,<) and x2+ x.”(Compare with the definition of an abstract basis in [l]). One of the arguments for the reasonability of the condition (5) is the following (naive) consideration:“If I know that x0 and x1 are approximations of an element x, then the pair (x0, x1) can be considered as an approximation of x, which contains only that information about x which is carried by the approximations x0 and XI. So from the point of view of information about xa couple (x0, x1) is the exact upper bound for x0 and XI.” The problem is that the pair (x0, x1) does not belong to the space of approximations X.

It is the aim of the present paper to present a mathematically correct realization of the idea described above and to show that for any a-space (= a basis for a domain, see Proposition 4 below) there exists a uniquely dejined bc-domain B and a homeomorphic embedding A: X-+ B with the properties of universality and minimality (the exact