The closure properties on real numbers under limits and computable operators

In e'ective analysis, various classes of real numbers are discussed. For example, the classes of computable, semi-computable, weakly computable, recursively approximable real numbers, etc. All these classes correspond to some kind of (weak) computability of the real numbers. In this paper we discuss mathematical closure properties of these classes under the limit, e'ective limit and computable function. Among others, we show that the class of weakly computable real numbers is not closed under e'ective limit and partial computable functions while the class of recursively approximable real numbers is closed under e'ective limit and partial computable functions. c © 2002 Elsevier Science B.V. All rights reserved.