Monafshefte flit ~h. Math. 83, 223--251 (1977) Malhemalik 9 by Springer-Verlag 1977 t~ber Distanzfunktionen mit Werten in angeordneten Halbgruppen Von Hans-Christian Reichel und Wolfgang Ruppert, Wien (Eingegangen am 7. Juli 1976) Herrn Prof. Dr. E. Hlawka zum 60. Geburtstag gewidmet* Abstract On Distance-Functions With Values in Linearly Ordered Semigroups. This paper presents a general investigation of the relations between struc- tural properties of a totally ordered abelian semigroup S and the properties of various "topological" structures, such as topologies, bitopologies and (semi-)uniformities on a space X induced by S-valued distance functions d: X • X -~S satisfying d(x,y) = 0 iff x = y and the triangular inequality d(x,z)<~d(x,y)+d(y,z), for all x,y,z~X. Since a linearly ordered abelian semigroup S need not be a topologieal semigroup with respect to its order topology we have to consider two cases: the case where addition in S is continuous at 0ES, and the case where it is not. For both cases, we state several metrization theorems, examples and applications. In this connection, we are also concerned with some special basis-properties of topological spaces. Closely connected is the program stated by A~EXAND~OF~-Bos~B~ (amongst others) to investigate to what extent countability inherent in metrization theory can be replaced by order-theoretic properties. -- Distin- guishing between symmetric and not necessarily symmetric distances d S we obtain a theory containing the theory of ~%-metrics and (t%-) quasimetrics. As far as it concerns not necessarily syrm~etric distances d on X, it seems ade- quate to study the bitopological structure (re, vr) induced on X by d and the "inverse" distance d -1 respectively. This is done in w4 where, in this respect, we also generalize a well-known theorem of SIo~ and ZET,~ER. Einleitung Die Struktur eines metrischen Raumes X wird dutch eine ,,Distanz", d. i. eine Abbildung d: X • X--> ~+, in die Halbgruppe der nicht negativen reellen Zahlen gegeben, wobei d gewisse ,,metri- * Die ersten Anrogungen zur grunds/itzliehen Problemstellung erhielt der erste Autor in einem Seminar fiber Bewertungstheorie und Analysis in bewerteten KSrpern, des unter der Leitung Herrn Prof. Dr. E. Iff~AWKAS am 1Vfathematischen Institut der Universit/~t Wien abgehalten wurde.