Abstract. This is a complete characterization of all possible simultaneous relations between the clones of uniformly continuous maps of two metric spaces and the respective clones of their continuous maps, in terms of the equality, isomorphism and elementary equivalence of their initial clone segments. In conjunction with earlier results, the apparatus introduced here gives a full characterization of the equality, isomorphism and elementary equivalence of clone segments for two topological spaces and their various lower and upper modifications, and a similar characterization of the segments of centralizer clones for two algebras with at least three non-nullary operations and their respective reducts.

AMS Subject Classification (1991): 54C05, 08C05

Keyword(s): clone, clone segment, finitary algebraic theory, subtheory, functors preserving finite products, algebras and their reducts, categories of universal algebras, categories of uniform or topological spaces, topological modifications, equality of clone segments, isomorphism of clone segments, elementary equivalence of clone segments

Received May 9, 2001, and in revised form April 7, 2002. (Registered under 2859/2009.)