The Weierstrass type results of Gajek and Zagrodny are not in general retainable in the precise context. Our first aim in this exposition is to show that a completion of the imposed conditions may be offered so that these results be true. As a second aim, alternate proofs of the statements in question are performed, via ordering principles comparable with the one in Brezis and Browder. | Turk J Math 30 (2006) , 385 – 401. ¨ ITAK ˙ c TUB Remarks About Some Weierstrass Type Results Mihai Turinici Abstract The Weierstrass type results of Gajek and Zagrodny [7] are not in general retainable in the precise context. Our first aim in this exposition is to show that a completion of the imposed conditions may be offered so that these results be true. As a second aim, alternate proofs of the statements in question are performed, via ordering principles comparable with the one in Brezis and Browder [3]. Key words and phrases: Relation, maximal element, extremal value, monotonically semicontinuous function, compact metric space, Brezis-Browder ordering principle, countably ordered structure. 1. Introduction Let M be some nonempty set. By a relation over it we mean any (nonempty) part S ⊆ M × M ; usually, we declare that (x, y) ∈ S is identical with xSy. For each n ≥ 2 denote S n = the n-th relational power of S: x(S n )y iff x = u1 ⊥ . ⊥ un = y (in the sense: ui ⊥ ui+1 , ∀i ∈ {1, ., n − 1}), for some u1 , ., un ∈ M . Further, put I := {(x, x); x ∈ M } (the diagonal of M ); and S −1 :=the (relational) inverse of S (introduced as: x(S −1 )y iff ySx). The relation S will be termed (a) reflexive, if I ⊆ S; (b) transitive, provided S 2 ⊆ S; (c) irreflexive if I ∩S = ∅; (d) antisymmetric when S∩S −1 ⊆ I; (e) quasi-order, provided it is reflexive and transitive; and (f) order, when it is 2000 AMS Mathematics Subject Classification: Primary 49J27; Secondary 49J53. 385 TURINICI antisymmetric and quasi-order. Returning to the general case, denote Sb = ∪{S n ; n ≥ 1} b These are, respectively, a transitive relation a quasi-order (where S 1 = S), Se = I ∪ S. including S and minimal with such properties; we shall refer to them as the transitive relation (respectively, quasi-order) induced by S. Finally, let R be another relation over M ; and V some nonempty part of M . We say that z ∈ V is (S, R)-maximal on V if w ∈ V and zSw imply wRz. () Note that,