A density result in vector optimization

The study of vector optimization problems has recently been a rapidly growing area of research. See, for example, [1–5] and the references mentioned therein. In this paper we study a class of vector minimization problems on a complete metric space such that all its bounded closed subsets are compact. This class of problems is associated with a complete metric space of continuous vector functions defined below. For each F from we denote by v(F) the set of all minimal elements of the image F(X)= {F(x) : x ∈ X}. In this paper we will study the sets v(F) with F ∈ . It is clear that for a minimization problem with only one criteria the set of minimal values is a singleton. In the present paper we will show that the subspace of all F ∈ with nonclosed sets v(F) is dense in . Therefore in general the sets v(F), F ∈ can be rather complicated. In this paper we use the convention that ∞/∞= 1 and denote by Card(E) the cardinality of the set E. Let R be the set of real numbers and let n be a natural number. Consider the finitedimensional space Rn with the norm