ALGEBRAIC GENERALIZATION OF THE TOPOLOGICAL THEOREMS OF BOLZANO AND WEIERSTRASS

Abstract

The couple structure—morphism as well as the couple category—covariant functor can be expressed in the form of the couple functional structure—homomorphism. These couples seem appropriate for the generalization and unification of apparently different classical results, for example, for the following couples: topological space—continuous function, uniform space—uniformly continuous function, ring—homomorphism, etc. An abstract theorem for functional structures contains particular cases: (1) the well-known Bolzano's theorem on the preservation of connexity by continuous surjections, (2) an obvious result for uniform spaces, and (3) the theorem on the one-to-one correspondence between the prime ideals of a ring B and the prime ideals of a ring A containing the kernel of a homomorphism of A onto B. The introduction of the notion of dimension of quasi-compactness allows showing that the well-known Weierstrass theorem on the preservation of quasi-compactness by continuous surjections and the algebraic statement that linear mappings do not increase linear dimension are all particular cases of the same abstract theorem for functional structures, asserting that, under certain rather general conditions, homomorphisms among functional structures do not increase the dimension of quasi-compactness.