tailieunhanh - Remarks about some weierstrass type results

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,

• Toán học
• Maximal element
• Extremal value
• Monotonically semicontinuous function
• Compact metric space
• Brezis Browder ordering principle
• Countably ordered structure