MARCINKIEWICZ CENTENARY VOLUME BANACH CENTER PUBLICATIONS, VOLUME 95 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2011 DEDEKIND CUTS IN C (X) NICOLAE DĂNEŢ Technical University of Civil Engineering of Bucharest Department of Mathematics and Computer Science 124, Lacul Tei Blvd., Bucharest, Romania E-mail: ndanet@cfdp.utcb.ro Abstract. The aim of this paper is to show that every Hausdorff continuous interval-valued function on a completely regular topological space X corresponds to a Dedekind cut in C(X) and conversely. 1. Introduction. The set R of real numbers can be constructed starting from the ra- tional set Q by the method developed by R. Dedekind in 1858 and which today is called “the order completion by Dedekind cuts” (see [9] or [16], pp. 17–21). In 1937, using the model of Dedekind cuts, H. M. MacNeille showed how the order completion of any par- tially ordered set can be obtained [15] (see also [14] or [19]). This completion is called Dedekind–MacNeille completion, Dedekind order completion or Dedekind completion for short. The monographs [14] and [19] contain not only the construction of the Dedekind completion of a partially ordered set, but also the construction of the Dedekind comple- tion of an Archimedean vector lattice. It is well known that the set C(X) of all real-valued continuous functions on a topolog- ical space X is an ordered set which is not Dedekind complete ([14], p. 125). R. P. Dilworth was the first who tried to obtain the Dedekind completion of C(X) by using MacNeille’s construction. In 1950 ([10], Theorem 4.1) he proved that the Dedekind completion of C b (X), the set of all real-valued bounded continuous functions on a completely regular topological space X, is order isomorphic with the lattice of all normal upper semicontinu- ous functions on X. (See Section 3 for the definition of normal semicontinuous functions.) To obtain this result Dilworth showed first that every normal subset of C b (X) corresponds to a normal upper semicontinuous function and conversely. (See Section 2 for the defini- tion of normal sets.) 2010 Mathematics Subject Classification : Primary 54C60, 26E25; Secondary 06B23. Key words and phrases : Hausdorff continuous interval-valued functions, Dedekind cuts, Dedekind order completion. The paper is in final form and no version of it will be published elsewhere. DOI: 10.4064/bc95-0-16 [287] c  Instytut Matematyczny PAN, 2011