0. Introductiori
Let X be a nonempty set and let s be a quasi-ordering (that is, a reflexive and transitive relation) on X. Given the function p : S -R, let us call the point z in S, cp-rnaximul when z zw implies cp(z) =cp (zu). A basic result about the existence of such elements is the 1976 BREZIS-BROWDER ordering principle [3] which, in a convenient manner, can be stated as follows.
Theorem 1. [S] Assume that (i) euch ascending sequence (x,JnEAY Iias ccn upper bound (ii) cp i s decreasing and bounded below.
T h e n , to each x E X there corresponds u cp-maximal element z E S with x 2 i3:
This principle, extending the one of EKELAND [ 6 ] and having a number of interesting applications to convex as well as nonconvex analysis (as the above references show) has been generalized in 1982 by ALTMAN [11 and TURINICI [18]. It is our first aim in the present exposition to get a common extension of these results in terms of (real) psezdometrics on X and, further, to clarify its relationships with another statements in this area due to GALVIN [7] (see Corollary 2 below) and the author [20, Theorem 21 Secondly, starting from the fact that, with respect to a certain structural condition of denumerable type the range of the ambient pseudometric is, in fact, unessential to the substance of the argument, a transfinite version of this ordering principle is formulated, as particular cases of i t being the abstract variant of BRBNDSTED'S result [4] obtained in V ~Y I
[21] the vector extension of the variational EKELAND'S principle (see the above reference) due to NEMETH [13, Proposition 2.31 as well as the locally convex generalization of the CARISTI fixed point theorem [5] given by ISAC [9]. Some further considerations about these problems will be developed elsewhere.