Roundings and Approximations in Ordered Sets

Abstract

The topological aspect of rounding was introduced by the author in an earlier article [1] and applied to the theory of a rounded algebraic law of composition. For example common rounding in ℝ as well as interval arithmetic on ℝ are covered by this general theory. In the present paper the development of the topological theory is continued by introducing the notions of adherence, co-adherence, and limit of a rounding, by a uniform topologization of the Space of filter- and ideal-basis on an ordered set with a rounding, by giving a general contraction principle, by a comparison of two roundings on the same ordered set, and by extending the theory of consistency and stability of approximations to mapping problems.