On completeness of quasi-pseudometric spaces

The notion of completeness in metric spaces and that of completing a metric space are traditionally discussed in terms of Cauchy sequences. The main reason being that this concept deals precisely with the issue of convergence of sequences in the sense that every convergent sequence is a Cauchy sequence. The paper deals with completion in a setting that avoids explicit reference to Cauchy sequences. First we recall some definitions which will be used in the paper, see also [3, 5, 9]. Let X be a set and let d : X ×X → [0,∞) be a function such that for all x, y,z ∈ X , (i) d(x,x) = 0, (ii) d(x, y) ≤ d(x,z) +d(z, y). Then d is called a quasi-pseudometric. If d is a quasi-pseudometric on X , then its conjugate denoted by d−1 on X is such that d−1(x, y) = d(y,x) for all x, y ∈ X . Certainly d−1 is a quasi-pseudometric. Let d∗ = d∨ d−1. Then d∗ is also quasi-pseudometric on X . If a quasi-pseudometric d on X satisfies d(x, y) = d(y,x) for all x, y ∈ X in addition to (i) and (ii), then d is called pseudometric. A pseudometric d that satisfies d(x, y) = 0 if and only if x = y is called a metric. Now if d is a quasi-pseudometric such that d(x, y) + d(y,x) > 0 for all x = y, then d is said to separate points in X . The set X equipped with a quasi-pseudometric d is said to be bounded if its diameter δ(X) = sup{d(x, y) : x, y ∈ X} exists. The open ball with centre x ∈ X and radius r > 0 is B(x,r) = {y ∈ X : d(x, y) < r}, and by τ(d) we denote the topology on X induced by d. We will say that a sequence {xn} in (X ,d) converges to x ∈ X if limn d(x,xn) = 0. When d is a separating pseudometric, then it is a metric, that is, the topology τ(d) is Hausdorff. By analogy with the topological case, in which a T2 space is said to be absolutely closed (H-closed) when it is closed in every T2 space that contains it, we recall the following.