# A maximality principle on ordered metric spaces

Se presenta un resultado sobre existencia de elementos maximales en espacios métricos ordenados el cual no requiere para su demostración el lemma de Zorn, basta usar el axioma de selecciones dependientes. Como aplicación se obtienen generalizaciones de resultados de Brezisy Browder, Bishop y Phelps, Ekeland y otros, A maximal element result on a class of order complete metric spaces, as well as an order type extension of a  Bishop-Phelps-Br

Tipo de documento: Artículo - Article

Palabras clave: Elementos maximales; espacios métricos; demostración; lemma de Zorn; axioma; Brezisy Browder, Bishop y Phelps, Ekeland y otros., Maximal elements, metric spaces, demonstration, Zorns lemma, axiom, Brezisy Browder, Bishop and Phelps, Ekeland

Source: http://www.bdigital.unal.edu.co

## Teaser

Re.v,uta.
Colornbian.a de.
Mllte.rna-uc.a.o VoL XVI (1982) pag~.
115 - 124 A MAXIMALITY PRINCIPLE ON ORDERED METRIC SPACES by Mihai TURINICI RESUMEN. Se presenta un resultado sobre existencia de elementos maximales en espacios metricos ordenados el cual no requiere para su demostracion el lemma de Zorn, basta usar el axioma de selecciones dependientes.
Como aplicacion se obtienen generalizaciones de resultados de Brezisy Browder, Bishop y Phelps, Ekeland y otros. ABSTRACT. A maximal element result on a class of order complete metric spaces, as well as an order type extension of a Bishop-Phelps-Br¢ndsted theorem about semlcontinuous functions are presented. A fundamental elements brated in an arbitrary Zorns theorem subsequent ambient result variants. ordered tures, making speaking, that may be formulated (partially) ordered (see, e.g., J.Kelley However, about maximal set is the cele- [10,p.33]) in many concrete situations, set is endowed with some sypplementary the above maximality unnecessary. principle and its the struc- to be, technically It is the main aim of the present note 115 to illustrate this assertion in case of an additional metric structure.
Moreover, it should be noted that, since in our reasonings only an ordinary induction argument is required, our maximal element result (stated below) might be considered at the same time as a metric version of a general ordering principle due to H.
Brezis and F.E.
Ekeland [8] ) . Let (X,d) be a metric space and let ~ be an o~d~g on X (i.e., a reflexive, antisymmetric and transtitive relation on X).
For every xEX, let X(x,~) denote the set of all YE:X with sequence (xn : n E IN) C X is said to be mOl1o-tone.
if and only if x .
~ x.
whenever i ~ j, i, j EN, and a.6 ymp-totic.
if x ~ y. A J l and only if d Cx ,x 1) a as n 00.
A subset YCX is called n n o~d~-cto~e.d if and only if for every monotone sequence (x n : nE:lN)cY we have x...