Two illustrating examples for comparison of uniform and proximal spaces using relators

Abstract

. We introduce (generalized) proximities in the same way as (gen-eralized) uniformities in paper of Weil. We prove the equivalence of our new definitions with classical ones. Using these analog definitions, we compare the properties of (generalized) proximities and (generalized) uniformities. The main parts of this paper are examples of an ( 𝑋, ℛ ) relator space such that ℛ # is uniformly (and proximally) transitive, but neither ℛ nor ℛ Φ is proximally (or uniformly) transitive. For this, we summarize the essential properties of relators, using their theory from earlier works of Á. Száz.