On Pseudometric Spaces 1

The terminology and notation used here have been introduced in the following articles: [9], [4], [13], [12], [10], [8], [2], [3], [1], [14], [7], [11], [5], and [6]. Let M be a metric structure, and let x, y be elements of the carrier of M . The predicate x ≈ y is defined by: (Def.1) ρ(x, y) = 0. Let M be a metric structure, and let x be an element of the carrier of M . The functor x yielding a subset of the carrier of M is defined as follows: (Def.2) x = {y : x ≈ y}, where y ranges over elements of the carrier of M . One can prove the following proposition (2) For every M being a metric structure and for every element x of the carrier of M holds x = {y : x ≈ y}, where y ranges over elements of the carrier of M . Let M be a metric structure. A subset of the carrier of M is called a equivalence class of M if: (Def.3) there exists an element x of the carrier of M such that it = x . Next we state a number of propositions: (4) For every pseudo metric space M and for every element x of the carrier of M holds x ≈ x. (5) For every pseudo metric space M and for all elements x, y of the carrier of M such that x ≈ y holds y ≈ x.