Comp. Appl. Math. (2013) 32:459–470 DOI 10.1007/s40314-013-0080-0 Interval metrics, topology and continuous functions Fagner Santana · Regivan Santiago Received: 24 December 2012 / Revised: 15 May 2013 / Accepted: 17 May 2013 / Published online: 10 October 2013 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2013 Abstract This paper describes a generalization of the mathematical concept of metric, in which the distance between two points is a closed interval. This generalization is called interval metric and it gives meaning to the phrase “the distance between x and y ranges in the interval [a, b]”. We show that the resulting topology is similar to those generated by usual metrics. Finally, a specific interval metric topology is proposed on the set of closed intervals and compared with Moore and Scott topologies. Keywords Interval metrics · Interval representation · KM-metrics Mathematics Subject Classification (2000) Primary 06B10; Secondary 06D05 1 Introduction Interval mathematics (IM) is a technique which arose in the 1950s to deal with numerical errors. According to Hayes (2003): “It does nothing directly to improve the accuracy of calculations, but for every number, it provides a certificate of accuracy—or the lack of it. The result of an ordinary (non-interval) computation is a single number, a point on the real line, which lies at some unknown distance from the true answer. An interval computation Communicated by Renata Hax Reiser Sander. F. Santana Department of Mathematics, Universidade Federal do Rio Grande do Norte, UFRN, 59072-970 Natal, Brazil e-mail: fagner@ccet.ufrn.br R. Santiago (B ) Group of Logic, Language, Information, Theory and Application, LoLITA, Department of Informatics and Applied Mathematics, DIMAp, Universidade Federal do Rio Grande do Norte, UFRN, 59072-970 Natal, Brazil e-mail: regivan@dimap.ufrn.br 123