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On Completeness of Quasi-Pseudometric Spaces ON COMPLETENESS OF QUASI-PSEUDOMETRIC SPACES SEITHUTI P. MOSHOKOA Received 4 July 2004 and in revised form 21 April 2005 We discuss completeness in terms of a notion of absolute closure. This will be done in the context of separated quasi-pseudometric spaces and bitopological spaces. The notion is equivalent to the classical notion of completeness when restricted to the class of metric spaces. 1. Introduction and preliminaries 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.Ifd is a quasi-pseudometric on X, then its con- jugate denoted by d−1 on X is such that d−1(x, y) = d(y,x)forallx, y ∈ X.Certainlyd−1 is a quasi-pseudometric. Let d∗ = d ∨ d−1.Thend∗ is also quasi-pseudometric on X.If a quasi-pseudometric d on X satisfies d(x, y) = d(y,x)forallx, y ∈ X in addition to (i) and (ii), then d is called pseudometric.Apseudometricd that satisfies d(x, y) = 0ifand only if x = y is called a metric. Now if d is a quasi-pseudometric such that d(x, y)+d(y,x) > 0forallx = y,thend 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>0isB(x,r) ={y ∈ X : d(x, y) <r},andbyτ(d) we denote the topology on X induced by d. We will say that a sequence {xn} in (X,d)convergestox ∈ X if limn d(x,xn) = 0. When d is a separating pseudometric, then it is a metric, that is, the topology τ(d)isHausdorff. 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. Copyright © 2005 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences 2005:18 (2005) 2933–2943 DOI: 10.1155/IJMMS.2005.2933 2934 Extending quasi-pseudometric spaces Definition 1.1 (see [6]). A metric space (X,d)isabsolutelyclosedifitsimageisaclosed subset of any space in which it can be embedded. Theorem 1.2 (see [6]). The metric space (X,d) is complete if and only if X is absolutely closed. Since the paper discusses completeness in spaces more general than metric spaces we provide a generalization of Definition 1.1 by considering a space (X,d), where d can be one of the functions we defined above, namely, a metric, a pseudometric, or a quasi- pseudometric on X.Byᏹ we will denote a class of spaces. In fact, we denote by ᏹm the class of all metric spaces, by ᏹp the class of all pseudometric spaces, and by ᏹq the class of all quasi-pseudometric spaces. The class ᏹ to be considered in each context will be explicitly mentioned. Firstly, we provide the following. Definition 1.3. The space (X,d)isabsolutelyclosedinᏹ if (X,d)isclosedineveryspace ¯ ¯ ¯ (X¯,d) that contains it, that is, (X¯,d) ∈ ᏹ and d|X×X = d. When this is the case we will simply say that (X,d)isᏹ-absolutely closed. So, Theorem 1.2 becomes as follows. Theorem 1.4. Let (X,d) be a metric space. Then (X,d) is complete if and only if (X,d) is ᏹm-absolutely closed. The paper [4] discusses a topological problem of characterizing the type of absolute “closure” properties possessed by compact spaces. In the paper, we discuss an analogous problem and completeness for the pseudometric spaces and quasi-pseudometric spaces by constructing explicit extensions for these spaces. 2. Completeness without Cauchy sequences The problem of completing a space is one of the classical extension problems in analysis and topology. Intuitively, a complete metric space is one in which every sequence that “should” converge does converge, in other words it is a space which cannot be extended as far as convergence of its sequences is concerned. We introduce the following definition and show that it provides us with an equivalent notion of the classical Cauchy-sequence completeness in the cases of a metric space and pseudometric space. Definition 2.1. Let (X,d)beaspace.Then(X,d)issaidtobeᏹ-convergence complete if ¯ ¯ for every space (X¯,d) ∈ ᏹ such that (X,d) ⊆ (X¯,d), a sequence {xn} in X that converges to a point x¯ in X¯ also converges to some point x in X. Then the following result is immediate. Proposition 2.2. Let (X,d) be a metric space. The following conditions are equivalent: (i) X is complete, (ii) X is ᏹm-convergence complete, (iii) X is ᏹm-absolutely closed. Seithuti P. Moshokoa 2935 Proof. (i)⇔(iii) follows by Theorem 1.4. Next we see that (ii)⇔(i) is clear, because in a metric space (X,d) a convergent sequence is Cauchy. We give a simple example. Example 2.3. Complete pseudometric spaces are ᏹp-convergence complete spaces, because in a pseudometric space a convergent sequence is a Cauchy sequence and a Cauchy se- quence in a complete pseudometric space has a limit. NowwegiveananalogueofProposition 2.2 in the context of pseudometric spaces. These spaces are not Hausdorff andwehavethefollowing. Proposition 2.4. Let (X,d) be a pseudometric space. Then the following conditions are equivalent: (i) X is complete, (ii) X is ᏹp-convergence complete. Observe that Proposition 2.2 does not hold for quasi-pseudometric spaces. These spaces are not symmetric and therefore there are several inequivalent notions of a Cauchy sequence which lead to different notions of a complete space, see [7, 8]forexample.In the next section we will examine the analogue of Proposition 2.2 in nonsymmetric spaces. In particular we discuss ᏹ-convergence completeness in quasi-pseudometric spaces. 3. Completeness of quasi-pseudometric spaces We start this section by further discussing extensions of spaces, and generalize our results to extensions of bounded quasi-pseudometric spaces. We first consider the following. Proposition 3.1. Let (X,d) be a pseudometric space. If (X,d) is ᏹq-convergence complete, then (X,d) is complete. ¯ Proof. We consider an arbitrary Cauchy sequence {xn} in X.If{xn} converges in (X¯,d) ∈ ᏹq that contains (X,d), then it also converges in X, since X is ᏹq-convergence complete, that is, every Cauchy sequence in X converges in X.Hence(X,d)iscomplete. When (X,d)inProposition 3.1 is a metric space, we get the following. Corollary 3.2. Let (X,d) be a metric space. If (X,d) is ᏹq-convergence complete, then (X,d) is complete. The following example shows that R with the usual metric is not ᏹq-convergence complete. Example 3.3. Let X = R with the usual metric. The sequence {(−1)n} in X is not conver- gent in X.LetX¯ = R ∪{∗}.Defined¯ : X¯ × X¯ → [0,∞)byd¯(x, y) =|x − y|,ifx, y ∈ R, d¯(∗,∗) = 0, −x +1 ifx<−1, −1 − x if x<−1, ¯ ∗ = ¯ ∗ = d(x, ) 2if− 1 ≤ x ≤ 1, d( ,x) 0if− 1 ≤ x ≤ 1, (3.1) x +1 ifx>1, x − 1ifx>1. 2936 Extending quasi-pseudometric spaces ¯ ¯ Then (X¯,d) is a quasi-pseudometric space that contains (X,d). Now d(∗,xn) = 0for ¯ n all n ∈ N,henced(∗,xn) → 0asn →∞,wherexn = (−1) . It follows that a sequence {xn} converges in X¯ but not in X. Therefore R is not ᏹq-convergence complete. Since X is complete when equipped with the usual metric, we see that the converse of Proposition 3.1 does not hold. Generally, we show that if a bounded quasi-pseudometric space (X,d)isclosedin every quasi-pseudometric space (X¯,d¯) that properly contains (X,d), then X is an empty set: in strong contrast with the metric space case. Theorem 3.4. Let (X,d) be a quasi-pseudometric space such that d is bounded and sepa- rates points in X. Then (X,d) is ᏹq-absolutely closed if and only if X is the empty set. Proof. Suppose that X is an empty set. Then every bounded quasi-pseudometric space (X¯,d¯) contains an empty set as a closed subset, hence we are done. Conversely, suppose that X is not an empty set. We construct a quasi-pseudometric space that contains X in the following way: for a point b/∈ X,letX¯ ={b}∪X.Defined¯: X¯ × X¯ → [0,∞)by d(x, y)ifx, y ∈ X, ¯ = d(x, y) 0ifx = b and y ∈ X¯, (3.2) M if x ∈ X and y = b, where M = δ(X). Now d¯(x,x) = 0forallx ∈ X¯ trivially holds, also the inequality d¯(u,v) ≤ d¯(u,w)+d¯(w,v) certainly holds for every u,v,w ∈ X¯.Henced¯ is a quasi-pseudometric and X ⊂ X¯. Now since X is nonempty, let a ∈ X.Foranyr>0, we have d(b,a) = 0 < r.Thuseveryopenballaboutb has a nonempty intersection with X,inparticulara ∈ B(b,r) ∩ X.
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