aa r X i v : . [ m a t h . G M ] M a y COMPLETENESS IN QUASI-PSEUDOMETRIC SPACES
S. COBZAS¸
Abstract.
The aim of this paper is to discus the relations between various notions of sequentialcompleteness and the corresponding notions of completeness by nets or by filters in the settingof quasi-metric spaces. We propose a new definition of right K -Cauchy net in a quasi-metricspace for which the corresponding completeness is equivalent to the sequential completeness. Inthis way we complete some results of R. A. Stoltenberg, Proc. London Math. Soc. (1967),226–240, and V. Gregori and J. Ferrer, Proc. Lond. Math. Soc., III Ser., (1984), 36. Classification MSC 2020:
Key words: metric space, quasi-metric space, uniform space, quasi-uniform space, Cauchysequence, Cauchy net, Cauchy filter, completeness Introduction
For a mapping d : X × X → R on a set X consider the following conditions:(M1) d ( x, y ) > d ( x, x ) = 0;(M2) d ( x, y ) = d ( y, x );(M3) d ( x, z ) d ( x, y ) + d ( y, z );(M4) d ( x, y ) = 0 ⇒ x = y ;(M4 ′ ) d ( x, y ) = d ( y, x ) = 0 ⇒ x = y, for all x, y, z ∈ X. The mapping d is called a pseudometric if it satisfies (M1), (M2) and (M3) and a metric if itfurther satisfies (M4).The open and closed balls in a pseudometric space ( X, d ) are defined by B d ( x, r ) = { y ∈ X : d ( x, y ) < r } and B d [ x, r ] = { y ∈ X : d ( x, y ) r } , respectively.A filter on a set X is a nonempty family F of nonempty subsets of X satisfying the conditions(F1) F ⊆ G and F ∈ F ⇒ G ∈ F ;(F2) F ∩ G ∈ F for all F, G ∈ F . It is obvious that (F2) implies(F2 ′ ) F , . . . , F n ∈ F ⇒ F ∩ . . . ∩ F n ∈ F . for all n ∈ N and F , . . . , F n ∈ F .A base of a filter F is a subset B of F such that every F ∈ F contains a B ∈ B .A nonempty family B of nonempty subsets of X such that(BF1) ∀ B , B ∈ B , ∃ B ∈ B , B ⊆ B ∩ B . Date : June 8, 2020. generates a filter F ( B ) given by F ( B ) = { U ⊆ X : ∃ B ∈ B , B ⊆ U } . A family B satisfying (BF1) is called a filter base .A uniformity on a set X is a filter U on X × X such that(U1) ∆( X ) ⊆ U, ∀ U ∈ U ;(U2) ∀ U ∈ U , ∃ V ∈ U , such that V ◦ V ⊆ U , (U3) ∀ U ∈ U , U − ∈ U . where ∆( X ) = { ( x, x ) : x ∈ X } denotes the diagonal of X,M ◦ N = { ( x, z ) ∈ X × X : ∃ y ∈ X, ( x, y ) ∈ M and ( y, z ) ∈ N } , and M − = { ( y, x ) : ( x, y ) ∈ M } , for any M, N ⊆ X × X. The sets in U are called entourages. A base for a uniformity U is a base of the filter U .The composition V ◦ V is denoted sometimes simply by V . Since every entourage contains thediagonal ∆( X ) , the inclusion V ⊆ U implies V ⊆ U. For U ∈ U , x ∈ X and Z ⊆ X put U ( x ) = { y ∈ X : ( x, y ) ∈ U } and U [ Z ] = [ { U ( z ) : z ∈ Z } . A uniformity U generates a topology τ ( U ) on X for which the family of sets { U ( x ) : U ∈ U } is a base of neighborhoods of any point x ∈ X. A base for a uniformity U is any base of the filter U . The following characterization of basescan be found in Kelley [6]. Proposition 1.1.
A nonempty family B of subsets of a set X × X is a base of a uniformity U if and only if (B1) ∆( X ) ⊆ B for any B ∈ B ; (B2) ∀ B ∈ B , ∃ C ∈ B , C ⊆ B ; (B3) ∀ B , B ∈ B , ∃ C ∈ B , C ⊆ B ∩ B ; (B4) ∀ B ∈ B , ∃ C ∈ B , C ⊆ B − .The corresponding uniformity is given by U = { U ⊆ X × X : ∃ B ∈ B , B ⊆ U } . A subbase of a uniformity U is a family B ⊆ U such that any U ∈ U contains the intersectionof a finite family of sets in B . Remark 1.2.
It can be shown (see, e.g., Kelley [6]) that a nonempty family B of subsets of aset X × X is a subbase of a uniformity U if and only if it satisfies the conditions (B1), (B2) and(B4) from Proposition 1.1.In this case the corresponding uniformity is given by U = { U ⊆ X × X : ∃ n ∈ N , ∃ B , . . . , B n ∈ B , B ∩ · · · ∩ B n ⊆ U } . OMPLETENESS IN QUASI-PSEUDOMETRIC SPACES 3
Let (
X, d ) be a pseudometric space. Then the pseudometric d generates a topology τ d forwhich B d ( x, r ) , r > , is a base of neighborhoods for every x ∈ X .The pseudometric d generates also a uniform structure U d on X having as basis of entouragesthe sets U ε = { ( x, y ) ∈ X × X : d ( x, y ) < ε } , ε > . Since U ε ( x ) = B d ( x, ε ) , x ∈ X, ε > , it follows that the topology τ ( U d ) agrees with the topology τ d generated by the pseudometric d .A sequence ( x n ) in X is called Cauchy (or fundamental ) if for every ε > n ε ∈ N such that d ( x n , x m ) < ε for all m, n ∈ N with m, n > n ε , a condition written also as lim m,n →∞ d ( x m , x n ) = 0 . A sequence ( x n ) in a uniform space ( X, U ) is called U - Cauchy (or simply
Cauchy ) if for every U ∈ U there exists n ∈ N such that( x m , x n ) ∈ U for all m, n ∈ N with m, n > n ε . It is obvious that in the case of a pseudometric space (
X, d ) a sequence is Cauchy with respectto the pseudometric d if and only if it is Cauchy with respect to the uniformity U d .The Cauchyness of nets in pseudometric or in uniform spaces is defined by analogy with thatof sequences.A filter F in a uniform space ( X, U ) is called U - Cauchy (or simply
Cauchy ) if for every U ∈ U there exists F ∈ F such that F × F ⊆ U .
Definition 1.3.
A pseudometric space (
X, d ) is called complete if every Cauchy sequence in X converges. A uniform space ( X, U ) is called sequentially complete if every U -Cauchy sequencein X converges and complete if every U -Cauchy net in X converges (or, equivalently, if every U -Cauchy filter in X converges). Remark 1.4.
We can define the completeness of a subset Y of X by the condition that everyCauchy sequence in Y converges to some element of Y . A closed subset of a pseudometricspace is complete and a complete subset of a metric space is closed. A complete subset of apseudometric space need not be closed.The following result holds in the metric case. Theorem 1.5.
For a pseudometric space ( X, d ) the following conditions are equivalent. The metric space X is complete. Every Cauchy net in X is convergent. Every Cauchy filter in ( X, U d ) is convergent. An important result in metric spaces is Cantor characterization of completeness.
S. COBZAS¸
Theorem 1.6 (Cantor theorem) . A pseudometric space ( X, d ) is complete if and only if ev-ery descending sequence of nonempty closed subsets of X with diameters tending to zero hasnonempty intersection. This means that for any family F n , n ∈ N , of nonempty closed subsetsof X F ⊇ F ⊇ . . . and lim n →∞ diam( F n ) = 0 ⇒ ∞ \ n =1 F n = ∅ . If d is a metric then this intersection contains exactly one point. The diameter of a subset Y of a pseudometric space ( X, d ) is defined by(1.1) diam( Y ) = sup { d ( x, y ) : x, y ∈ Y } . Quasi-pseudometric and quasi-uniform spaces
Quasi-pseudometric spaces.
Dropping the symmetry condition (M2) in the definitionof a metric one obtains the notion of quasi-pseudometric, that is, a quasi-pseudometric on anarbitrary set X is a mapping d : X × X → R satisfying the conditions (M1) and (M3). If d satisfies further (M4 ′ ) then it called a quasi-metric . The pair ( X, d ) is called a quasi-pseudometricspace , respectively a quasi-metric space .The conjugate of the quasi-pseudometric d is the quasi-pseudometric ¯ d ( x, y ) = d ( y, x ) , x, y ∈ X. The mapping d s ( x, y ) = max { d ( x, y ) , ¯ d ( x, y ) } , x, y ∈ X, is a pseudometric on X which is ametric if and only if d is a quasi-metric.If ( X, d ) is a quasi-pseudometric space, then for x ∈ X and r > X bythe formulae B d ( x, r ) = { y ∈ X : d ( x, y ) < r } - the open ball, and B d [ x, r ] = { y ∈ X : d ( x, y ) r } - the closed ball.The topology τ d (or τ ( d )) of a quasi-pseudometric space ( X, d ) can be defined through thefamily V d ( x ) of neighborhoods of an arbitrary point x ∈ X : V ∈ V d ( x ) ⇐⇒ ∃ r > B d ( x, r ) ⊆ V ⇐⇒ ∃ r ′ > B d [ x, r ′ ] ⊆ V. The topological notions corresponding to d will be prefixed by d - (e.g. d -closure, d -open, etc).The convergence of a sequence ( x n ) to x with respect to τ d , called d -convergence and denotedby x n d −→ x, can be characterized in the following way(2.1) x n d −→ x ⇐⇒ d ( x, x n ) → . Also(2.2) x n ¯ d −→ x ⇐⇒ ¯ d ( x, x n ) → ⇐⇒ d ( x n , x ) → . As a space equipped with two topologies, τ d and τ ¯ d , a quasi-pseudometric space can be viewedas a bitopological space in the sense of Kelly [7]. Asymmetric normed spaces
Let X be a real vector space. A mapping p : X → R is called an asymmetric seminorm on X if In [4] the term “quasi-semimetric” is used instead of “quasi-pseudometric”
OMPLETENESS IN QUASI-PSEUDOMETRIC SPACES 5 (AN1) p ( x ) > p ( αx ) = αp ( x );(AN3) p ( x + y ) p ( x ) + p ( y ) , for all x, y ∈ X and α > . If, further, (AN4) p ( x ) = p ( − x ) = 0 ⇒ x = 0 , for all x ∈ X, then p is called an asymmetric norm .To an asymmetric seminorm p one associates a quasi-pseudometric d p given by d p ( x, y ) = p ( y − x ) , x, y ∈ X, which is a quasi-metric if p is an asymmetric norm. All the topological and metric notions inan asymmetric normed space are understood as those corresponding to this quasi-pseudometric d p (see [4]).The following topological properties are true for quasi-pseudometric spaces. Proposition 2.1 (see [4]) . If ( X, d ) is a quasi-pseudometric space, then the following hold. The ball B d ( x, r ) is d -open and the ball B d [ x, r ] is ¯ d -closed. The ball B d [ x, r ] need notbe d -closed. The topology d is T if and only if d is a quasi-metric.The topology d is T if and only if d ( x, y ) > for all x = y in X . For every fixed x ∈ X, the mapping d ( x, · ) : X → ( R , | · | ) is d -usc and ¯ d -lsc.For every fixed y ∈ X, the mapping d ( · , y ) : X → ( R , | · | ) is d -lsc and ¯ d -usc. Remark 2.2.
It is known that the topology τ d of a pseudometric space ( X, d ) is Hausdorff (or T ) if and only if d is a metric if and only if any sequence in X has at most one limit.The characterization of Hausdorff property of quasi-pseudometric spaces can also be given interms of uniqueness of the limits, as in the metric case: the topology of a quasi-pseudometricspace ( X, d ) is Hausdorff if and only if every sequence in X has at most one d -limit if and onlyif every sequence in X has at most one ¯ d -limit (see [17]).In the case of an asymmetric seminormed space there exists a characterization in terms of theasymmetric seminorm (see [4], Proposition 1.1.40).Recall that a topological space ( X, τ ) is called: • T if, for every pair of distinct points in X , at least one of them has a neighborhood notcontaining the other; • T if, for every pair of distinct points in X , each of them has a neighborhood notcontaining the other; • T (or Hausdorff ) if every two distinct points in X admit disjoint neighborhoods; • regular if, for every point x ∈ X and closed set A not containing x , there exist thedisjoint open sets U, V such that x ∈ U and A ⊆ V. Quasi-uniform spaces.
Again, the notion of quasi-uniform space is obtained by droppingthe symmetry condition (U3) from the definition of a uniform space, that is, a quasi-uniformity on a set X is a filter U in X × X satisfying the conditions (U1) and (U2). The sets in U arecalled entourages and the pair ( X, U ) is called a quasi-uniform space , as in the case of uniformspaces. S. COBZAS¸
As uniformities, a quasi-uniformity U generates a topology τ ( U ) on X in a similar way: thesets { U ( x ) : U ∈ U } form a base of neighborhoods of any point x ∈ X. The topology τ ( U ) is T if and only if T U is a partial order on X , and T if and only if T U = ∆( X ).The family of sets(2.3) U − = { U − : U ∈ U } is another quasi-uniformity on X called the quasi-uniformity conjugate to U . Also U ∪ U − isa subbase of a uniformity U s on X , called the associated uniformity to the quasi-uniformity U .It is the coarsest uniformity on X finer than both U and U − , U s = U ∨ U − . A basis for U s isformed by the sets { U ∩ U − : U ∈ U } . If (
X, d ) is a quasi-pseudometric space, then U ε = { ( x, y ) ∈ X × X : d ( x, y ) < ε } , ε > , is a basis for a quasi-uniformity U d on X. The family U − ε = { ( x, y ) ∈ X × X : d ( x, y ) ε } , ε > , generates the same quasi-uniformity. Since U ε ( x ) = B d ( x, ε ) and U − ε ( x ) = B d [ x, ε ], it followsthat the topologies generated by the quasi-pseudometric d and by the quasi-uniformity U d agree,i.e., τ d = τ ( U d ) . In this case U − d = U ¯ d and U sd = U d s . Cauchy sequences and sequential completeness in quasi-pseudometric andquasi-uniform spaces
In contrast to the case of metric or uniform spaces, completeness, total boundedness andcompactness look very different in quasi-metric and quasi-uniform spaces, due to the lack ofsymmetry of the distance. The present paper is concerned only with completeness. There areseveral notions of completeness in quasi-metric and quasi-uniform spaces, all agreeing with theusual notion of completeness in the case of metric or uniform spaces, each of them having itsadvantages and weaknesses.We introduce now some of these notions following [10] (see also [4]).
Definition 3.1.
A sequence ( x n ) in ( X, d ) is called • left (right) d -Cauchy if for every ε > x ∈ X and n ∈ N such that d ( x, x n ) < ε (respectively d ( x n , x ) < ε )for all n > n ; • d s -Cauchy if it is a Cauchy sequence is the pseudometric space ( X, d s ), that is for every ε > n ∈ N such that d s ( x n , x k ) < ε for all n, k > n , or, equivalently, d ( x n , x k ) < ε for all n, k > n ; • left (right) K -Cauchy if for every ε > n ∈ N such that d ( x k , x n ) < ε (respectively d ( x n , x k ) < ε )for all n, k ∈ N with n k n ; OMPLETENESS IN QUASI-PSEUDOMETRIC SPACES 7 • weakly left (right) K -Cauchy if for every ε > n ∈ N such that d ( x n , x n ) < ε (respectively d ( x n , x n ) < ε ) , for all n > n . Sometimes, to emphasize the quasi-pseudometric d, we shall say that a sequence is left d - K -Cauchy, etc.It seems that K in the definition of a left K -Cauchy sequence comes from Kelly [7] whoconsidered first this notion.Some remarks are in order. Remark 3.2 ([10]) . Let (
X, d ) be a quasi-pseudometric space.1. These notions are related in the following way: d s -Cauchy ⇒ left K -Cauchy ⇒ weakly left K -Cauchy ⇒ left d -Cauchy.The same implications hold for the corresponding right notions. No one of the aboveimplications is reversible.2. A sequence is left Cauchy (in some sense) with respect to d if and only if it is rightCauchy (in the same sense) with respect to ¯ d.
3. A sequence is d s -Cauchy if and only if it is both left and right d - K -Cauchy.4. A d -convergent sequence is left d -Cauchy and a ¯ d -convergent sequence is right d -Cauchy.For the other notions, a convergent sequence need not be Cauchy.5. If each convergent sequence in a regular quasi-metric space ( X, d ) admits a left K -Cauchysubsequence, then X is metrizable ([9]).We also mention the following simple properties of Cauchy sequences. Proposition 3.3 ([2, 11]) . Let ( x n ) be a left or right K -Cauchy sequence in a quasi-pseudometricspace ( X, d ) . If ( x n ) has a subsequence which is d -convergent to x, then ( x n ) is d -convergent to x. If ( x n ) has a subsequence which is ¯ d -convergent to x, then ( x n ) is ¯ d -convergent to x. If ( x n ) has a subsequence which is d s -convergent to x , then ( x n ) is d s -convergent to x . To each of these notions of Cauchy sequence corresponds two notions of sequential complete-ness, by asking that the corresponding Cauchy sequence be d -convergent or d s -convergent. Dueto the equivalence d -left Cauchy ⇐⇒ ¯ d -right Cauchy one obtains nothing new by asking that a d -left Cauchy sequence is ¯ d -convergent. For instance, the ¯ d -convergence of any left d - K -Cauchysequence is equivalent to the right K -completeness of the space ( X, ¯ d ) . Definition 3.4 ([10]) . A quasi-pseudometric space (
X, d ) is called: • sequentially d -complete if every d s -Cauchy sequence is d -convergent; • sequentially left d -complete if every left d -Cauchy sequence is d -convergent; • sequentially weakly left (right) K -complete if every weakly left (right) K -Cauchy sequenceis d -convergent; • sequentially left (right) K -complete if every left (right) K -Cauchy sequence is d -convergent; • sequentially left (right) Smyth complete if every left (right) K -Cauchy sequence is d s -convergent; • bicomplete if the associated pseudometric space ( X, d s ) is complete, i.e., every d s -Cauchysequence is d s -convergent. A bicomplete asymmetric normed space ( X, p ) is called a biBanach space.
S. COBZAS¸
As we noticed (see Remark 3.2.4), each d -convergent sequence is left d -Cauchy, but for each ofthe other notions there are examples of d -convergent sequences that are not Cauchy, which is amajor inconvenience. Another one is that a complete (in some sense) subspace of a quasi-metricspace need not be closed.The implications between these completeness notions are obtained by reversing the implica-tions between the corresponding notions of Cauchy sequence from Remark 3.2.1. Remark 3.5. (a) These notions of completeness are related in the following way:sequentially d -complete ⇒ sequentially weakly left K -complete ⇒ sequentially left K -complete ⇒ sequentially left d -complete.The same implications hold for the corresponding notions of right completeness.(b) sequentially left or right Smyth completeness implies bicompleteness.No one of the above implication is reversible (see [10]), excepting that between weakly leftand left K -sequential completeness, as it was surprisingly shown by Romaguera [12]. Proposition 3.6 ([12], Proposition 1) . A quasi-pseudometric space is sequentially weakly left K -complete if and only if it is sequentially left K -complete. A series P n x n in an asymmetric seminormed space ( X, p ) is called convergent if there exists x ∈ X such that x = lim n →∞ P nk =1 x k (i.e., lim n →∞ p ( P nk =1 x k − x ) = 0). The series P n x n is called absolutely convergent if P ∞ n =1 p ( x n ) < ∞ . It is well-known that a normed space iscomplete if and only if every absolutely convergent series is convergent. A similar result holdsin the asymmetric case too.
Proposition 3.7.
Let ( X, d ) be a quasi-pseudometric space. If a sequence ( x n ) in X satisfies P ∞ n =1 d ( x n , x n +1 ) < ∞ ( P ∞ n =1 d ( x n +1 , x n ) < ∞ ) , thenit is left (right) d - K -Cauchy. The quasi-pseudometric space ( X, d ) is sequentially left (right) d - K -complete if and onlyif every sequence ( x n ) in X satisfying P ∞ n =1 d ( x n , x n +1 ) < ∞ (resp. P ∞ n =1 d ( x n +1 , x n ) < ∞ ) is d -convergent. An asymmetric seminormed space ( X, p ) is sequentially left K -complete if and only ifevery absolutely convergent series is convergent. Cantor type results
Concerning Cantor-type characterizations of completeness in terms of descending sequencesof closed sets (the analog of Theorem 1.6) we mention the following result. The diameter of asubset A of a quasi-pseudometric space ( X, d ) is defined by(3.1) diam( A ) = sup { d ( x, y ) : x, y ∈ A } . It is clear that, as defined, the diameter is, in fact, the diameter with respect to the associatedpseudometric d s . Recall that a quasi-pseudometric space is called sequentially d -complete if every d s -Cauchy sequence is d -convergent (see Definition 3.4). Theorem 3.8 ([10], Theorem 10) . A quasi-pseudometric space ( X, d ) is sequentially d -completeif and only if each decreasing sequence F ⊇ F . . . of nonempty closed sets with diam( F n ) → as n → ∞ has nonempty intersection, which is a singleton if d is a quasi-metric. The following characterization of right K -completeness was obtained in [3], using a differentterminology. OMPLETENESS IN QUASI-PSEUDOMETRIC SPACES 9
Proposition 3.9.
A quasi-pseudometric space ( X, d ) is sequentially right K -complete if andonly if any decreasing sequence of closed ¯ d -balls B ¯ d [ x , r ] ⊇ B ¯ d [ x , r ] ⊇ . . . with lim n →∞ r n = 0 , has nonempty intersection.If the topology d is Hausdorff, then T ∞ n =1 B ¯ d [ x n , r n ] contains exactly one element. Completeness by nets and filters
The Cauchy properties of a net ( x i : i ∈ I ) in a quasi-pseudometric space ( X, d ) are definedby analogy with that of sequences, by replacing in Definition 3.1 the natural numbers with theelements of the directed set I .The situation is good for left Smyth completeness (see Definition 3.4). Proposition 4.1 ([13], Prop. 1) . For a quasi-metric space ( X, d ) the following are equivalent. Every left d - K -Cauchy sequence is d s -convergent. Every left d - K -Cauchy net is d s -convergent. A quasi-uniform space ( X, U ) is called bicomplete if ( X, U s ) is a complete uniform space. Thisnotion is useful and easy to handle, because one can appeal to well known results from the theoryof uniform spaces, but it is not appropriate for the study of the specific properties of quasi-uniform spaces, so one introduces adequate definitions, by analogy with quasi-pseudometricspaces. Definition 4.2.
Let ( X, U ) be a quasi-uniform space.A filter F on ( X, U ) is called: • left ( right ) U - Cauchy if for every U ∈ U there exists x ∈ X such that U ( x ) ∈ F (respectively U − ( x ) ∈ F ); • left ( right ) U - K - Cauchy if for every U ∈ U there exists F ∈ F such that U ( x ) ∈ F (resp. U − ( x ) ∈ F ) for all x ∈ F .A net ( x i : i ∈ I ) in ( X, U ) is called: • left U -Cauchy ( right U -Cauchy ) if for every U ∈ U there exists x ∈ X and i ∈ I such that( x, x i ) ∈ U (respectively ( x i , x ) ∈ U ) for all i > i ; • left U - K -Cauchy ( right U - K -Cauchy ) if(4.1) ∀ U ∈ U , ∃ i ∈ I, ∀ i, j ∈ I, i i j ⇒ ( x i , x j ) ∈ U (resp. ( x j , x i ) ∈ U .
The notions of left and right U - K -Cauchy filter were defined by Romaguera in [12].Observe that ( x j , x i ) ∈ U ⇐⇒ ( x i , x j ) ∈ U − , so that a filter is right U - K -Cauchy if and only if it is left U − - K -Cauchy. A similar remarkapplies to U -nets. Definition 4.3.
A quasi-uniform space ( X, U ) is called: • left U -complete by filters ( left K -complete by filters ) if every left U -Cauchy (respectively,left U - K -Cauchy) filter in X is τ ( U )-convergent; • left U -complete by nets ( left U - K -complete by nets ) if every left U -Cauchy (respectively,left U - K -Cauchy) net in X is τ ( U )-convergent; • Smyth left U - K -complete by nets if every left K -Cauchy net in X is U s -convergent.The notions of right completeness are defined similarly, by asking the τ ( U )-convergence of thecorresponding right Cauchy filter (or net) with respect to the topology τ ( U ) (or with respect to τ ( U s ) in the case of Smyth completeness). As we have mentioned in Introduction, in pseudometric spaces the sequential completeness isequivalent to the completeness defined in terms of filters, or of nets. Romaguera [12] proved asimilar result for the left K -completeness in quasi-pseudometric spaces. Remark 4.4.
In the case of a quasi-pseudometric space the considered notions take the followingform.A filter F in a quasi-pseudometric space ( X, d ) is called left K -Cauchy if it left U d - K -Cauchy.This is equivalent to the fact that for every ε > F ε ∈ F such that(4.2) ∀ x ∈ F ε , B d ( x, ε ) ∈ F . Also a net ( x i : i ∈ I ) is called left K - Cauchy if it is left U d - K -Cauchy or, equivalently, forevery ε > i ∈ I such that(4.3) ∀ i, j ∈ I, i i j ⇒ d ( x i , x j ) < ε . Proposition 4.5 ([12]) . For a quasi-pseudometric space ( X, d ) the following are equivalent. The space ( X, d ) is sequentially left K -complete. Every left K -Cauchy filter in X is d -convergent. Every left K -Cauchy net in X is d -convergent. In the case of left U d -completeness this equivalence does not hold in general. Proposition 4.6 (K¨unzi [8]) . A Hausdorff quasi-metric space ( X, d ) is sequentially left d -complete if and only if the associated quasi-uniform space ( X, U d ) is left U d -complete by filters. Right K -completeness in quasi-pseudometric spaces. It is strange that for the rightcompleteness the things look worse than for the left completeness.As remarked Stoltenberg [15, Example 2.4] a result similar to Proposition 4.5 does not holdfor right K -completeness: there exists a sequentially right K -complete T quasi-metric spacewhich is not right K -complete by nets. Actually, Stoltenberg [15] proved that the equivalenceholds for a more general definition of a right K -Cauchy net, see Proposition 4.15.An analog of Proposition 4.5 for right K -completeness can be obtained only under somesupplementary hypotheses on the quasi-pseudometric space X .A quasi-pseudometric space ( X, d ) is called R if for all x, y ∈ X, d -cl { x } 6 = d -cl { y } impliesthe existence of two disjoint d -open sets U, V such that x ∈ U and y ∈ V. Proposition 4.7 ([1]) . Let ( X, d ) be a quasi-pseudometric space. The following are true. If X is right K -complete by filters, then every right K -Cauchy net in X is convergent.In particular, every right K -complete by filters quasi-pseudometric space is sequentiallyright K -complete. If the quasi-pseudometric space ( X, d ) is R then X is right K -complete by filters if andonly if it is sequentially right K -complete. Stoltenberg’s example
As we have mentioned, Stoltenberg [15, Example 2.4] gave an example of a sequentially right K -complete T quasi-metric space which is not right K -complete by nets, which we shall presentnow.Denote by A the family of all countable subsets of the interval [0 , ]. For A ∈ A let A A = A, X Ak +1 = A ∪ (cid:26) , , . . . , k − k (cid:27) , k ∈ N , and X A ∞ = A ∪ (cid:26) k − k : k ∈ N (cid:27) = [ { X Ak : k ∈ N } . OMPLETENESS IN QUASI-PSEUDOMETRIC SPACES 11
Put S = (cid:8) X Ak : A ∈ A , k ∈ N ∪ {∞} (cid:9) and define d : S × S → [0 , ∞ ) by d ( X Ak , X Bj ) = A = B and k = j, A, B ∈ A , k, j ∈ N ∪ {∞} − j if X Bj $ X Ak , A, B ∈ A , k ∈ N ∪ {∞} , j ∈ N , . Proposition 4.8. ( S , d ) is a sequentially right K -complete T quasi-metric space which is notright K -complete by nets.Proof. The proof that d is a T quasi-metric on S is straightforward.I. ( S , d ) is sequentially right K - complete .Let ( X n ) n ∈ N be a right K -Cauchy sequence in S . Then there exists n ∈ N such that d ( X m , X n ) < m, n ∈ N with n n m . For i ∈ N = N ∪ { } let X n + i = X A i k i where A i ∈ A and k i ∈ N ∪ {∞} . Since d (cid:16) X A i +1 k i +1 , X A i k i (cid:17) < , it follows k i ∈ N for all i ∈ N . For 0 < ε < i ∈ N such that d ( X n + i +1 , X n + i ) < ε for all i > i , which means that 2 − k i = d (cid:16) X A i +1 k i +1 , X A i k i (cid:17) < ε for all i > i . This shows that lim i →∞ k i = ∞ . Let A = S { A i : i ∈ N } and X = X A ∞ . Then X A i k i $ X A ∞ , so that d ( X, X n + i ) = d (cid:16) X A ∞ , X A i k i (cid:17) = 2 − k i → i → ∞ . which shows that the sequence ( X n ) is d -convergent to X .II. The quasi-metric space ( S , d ) is not right K -complete by nets. Let S = { X Ak : A ∈ A , k ∈ N } ordered by X Y ⇐⇒ X ⊆ Y, for X, Y ∈ S . We have X Ai X Bj ⇐⇒ ( A ⊆ B and i j for X Ai , X Bj ∈ S , ( S , ) is directed and the mapping φ : S → S defined by φ ( X ) = X, X ∈ S , is a net in S . Let us show first that the net φ is right K -Cauchy. For ε > k ∈ N such that 2 − k < ε. For some C ∈ A , X Ck belongs to S and d ( X Aj , X Bi ) = 2 − i − k < ε for all X Aj , X Bi ∈ S with X Ck X Bi X Aj , X Aj = X Bi , showing that the net φ is right K -Cauchy. Let X = X Ck be an arbitrary element in S . We show that for every X Ai ∈ S there exists X Bj ∈ S with X Ai X Bj such that d ( X, X Bj ) = 1 , which will imply that the net φ is not d -convergent to X .Since C is a countable set, there exists x ∈ [0 , ] \ C. For an arbitrary X Ai ∈ S let B = A ∪ { x } . Then X Bi ∈ S , X Ai X Bi and X Ck * X Bj , so that, by the definition of the metric d , d ( X Ck , X Bi ) = 1 . (cid:3) Stoltenberg-Cauchy nets
Stoltenberg [15] also considered a more general definition of a right K -Cauchy net as a net( x i : i ∈ I ) satisfying the condition: for every ε > i ε ∈ I such that(4.4) d ( x i , x j ) < ε for all i, j > i ε with i (cid:10) j . Let us call such a net
Stoltenberg-Cauchy and
Stoltenberg completeness the completeness withrespect to Stoltenberg-Cauchy nets.It follows that, for this definition, d ( x j , x i ) < ε and d ( x j , x i ) < ε for all i, j > i ε with i ≁ j, where i ≁ j means that i, j are incomparable (that is, no one of the relations i j or j i holds). Gregori-Ferrer-Cauchy nets
Later, Gregori and Ferrer [5] found a gap in the proof of Theorem 2.5 from [15] and provideda counterexample to it, based on Example 2.4 of Stoltenberg (see Proposition 4.8).
Example 4.9 ([5]) . Let A , ( S , d ) be as in the preamble to Proposition 4.8 and I = N ∪ { a, b } ,where the set N is considered with the usual order and a, b are two distinct elements not belongingto N with k a, k b, for all k ∈ N ,a a, b b a b, b a. Consider two sets
A, B ∈ A with A $ B and let φ : I → S be given by φ ( k ) = X Ak , k ∈ N , φ ( a ) = X A ∞ , φ ( b ) = X B ∞ . Then the net φ is right Cauchy in the sense of (4.4) but not convergent in ( S , d ).Indeed, for 0 < ε < k ∈ N be such that 2 − k < ε. Since i a, i b, i > k , ∀ i ∈ N , k a b, k b a, it follows that the condition i (cid:10) j can hold for some i, j ∈ I, i, j > k , in the following cases:(a) i, j ∈ N , i, j > k , j < i ;(b) i = a, j ∈ N , j > k (c) i = b, j ∈ N , j > k In the case (a), X Aj $ X Ai and d ( φ ( i ) , φ ( j )) = d ( X Ai , X Aj ) = 2 − j − k < ε . In the case (b), X Aj $ X A ∞ and again OMPLETENESS IN QUASI-PSEUDOMETRIC SPACES 13 d ( φ ( a ) , φ ( j )) = d ( X A ∞ , X Aj ) = 2 − j − k < ε. The case (c) is similar to (b).To show that φ is not convergent let X ∈ S r { X B ∞ } . Then b > i for any i ∈ I and d ( X, φ ( b )) = d ( X, X B ∞ ) = 1 , so that φ does not converge to X . If X = X B ∞ , then a > i for any i ∈ I and d ( X B ∞ , φ ( a )) = d ( X B ∞ , X A ∞ ) = 1 . Gregori and Ferrer [5] proposed a new definition of a right K-Cauchy net, for which theequivalence to sequential completeness holds.
Definition 4.10.
A net ( x i : i ∈ I ) in a quasi-metric space ( X, d ) is called GF - Cauchy if oneof the following conditions holds:(a) for every maximal element j ∈ I the net ( x i ) converges to x j ;(b) I has no maximal elements and the net ( x i ) converges;(c) I has no maximal elements and the net ( x i ) satisfies the condition (4.4). Maximal elements and net convergence
For a better understanding of this definition we shall analyze the relations between maximalelements in a preordered set and the convergence of nets. Recall that in the definition of adirected set ( I, ) the relation is supposed to be only a preorder, i.e. reflexive and transitiveand not necessarily antireflexive (see [6]). Notice that some authors suppose that in the definitionof a directed set is a partial order (see, e.g., [16]). For a discussion of this matter see [14, § I, ) be a preordered set. An element j ∈ I is called: • strictly maximal if there is no i ∈ I r { j } with j i, or, equivalently,(4.5) j i ⇒ i = j, for every i ∈ I ; • maximal if(4.6) j i ⇒ i j, for every i ∈ I .
Remark 4.11.
Let ( I, ) be a preordered set.1. A strictly maximal element is maximal, and if is an order, then these notions areequivalent.Suppose now that the set I is further directed. Then the following hold.2. Every maximal element j of I is a maximum for I , i.e. i j for all i ∈ I .3. If j is a maximal element and j ′ ∈ I satisfies j j ′ , then j ′ is also a maximal element.4. (Uniqueness of the strictly maximal element) If j is a strictly maximal element, then j ′ = j for any maximal element j ′ of I . Proof.
1. These assertions are obvious.2. Indeed, suppose that j ∈ I satisfies (4.6). Then, for arbitrary i ∈ I , there exists i ′ ∈ I with i ′ > j, i. But j i ′ implies i ′ j and so i i ′ j. (We use the notation i > j, k for i > j ∧ i > k .)3. Let i ∈ I be such that j ′ i . Then j i and, by the maximality of j , i j j ′ .4. If j is strictly maximal and j ′ is a maximal element of I , then, by 2, j ′ ≤ j so that, by(4.6) applied to j ′ , j ≤ j ′ and so, by (4.5) applied to j , j ′ = j. (cid:3) We present now some remarks on maximal elements and net convergence.
Remark 4.12.
Let (
X, d ) be a quasi-metric space, ( I, ) a directed sets and ( x i : i ∈ I ) a netin X .1. If ( I, ) has a strictly maximal element j , then the net ( x i ) is convergent to x j .2.(a) If the net ( x i ) converges to x ∈ X , then d ( x, x j ) = 0 for every maximal element j of I .If the topology τ d is T then, further, x j = x. (b) If the net ( x i ) converges to x j and to x j ′ , where j, j ′ are maximal elements of I , then x j = x j ′ .(c) If I has maximal elements and, for some x ∈ X , x j = x for every maximal element j ,then the net ( x i ) converges to x . Proof.
1. For an arbitrary ε > i ε = j. Then i > j implies i = j , so that d ( x j , x i ) = d ( x j , x j ) = 0 < ε . ε > i ε ∈ I such that d ( x, x i ) < ε for all i > i ε . By Remark4.11.2, j > i ε for every maximal j , so that d ( x, x j ) < ε for all ε >
0, implying d ( x, x j ) = 0.If the topology τ d is T , then, by Proposition 2.1.2, x j = x .(b) By (a), d ( x j , x j ′ ) = 0 and d ( x j ′ , x j ) = 0, so that x j = x j ′ .(c) Let x ∈ X be such that x j = x for every maximal element j of I and let j be a fixedmaximal element of I . For any ε > i ε = j . Then, by Remark 4.11.3, any i ∈ I such that i > j is also a maximal element of I , so that x i = x and d ( x, x i ) = 0 < ε. (cid:3) Let us say that a quasi-metric space (
X, d ) is
GF-complete if every GF-Cauchy net (i.e.satisfying the conditions (a),(b),(c) from Definition 4.10) is convergent. Remark that, with thisdefinition, condition (b) becomes tautological and so superfluous, so it suffices to ask that everynet satisfying (a) and (c) be convergent.By Remarks 4.11.1 and 4.12.1, (a) always holds if is an order, so that, in this case, a netsatisfying condition (c) is a GF-Cauchy net and so GF-completeness agrees with that given byStoltenberg. Strongly Stoltenberg-Cauchy nets
In order to avoid the shortcomings of the preorder relation, as, for instance, those put inevidence by Example 4.9, we propose the following definition.
Definition 4.13.
A net ( x i : i ∈ I ) in a quasi-metric space ( X, d ) is called strongly Stoltenberg-Cauchy if for every ε > i ε ∈ I such that, for all i, j > i ε ,(4.7) ( j i ∨ i ≁ j ) ⇒ d ( x i , x j ) < ε . We present now some remarks on the relations of this notion with other notions of Cauchy net(Stoltenberg and GF), as well as the relations between the corresponding completeness notions.It is obvious that in the case of a sequence ( x k ) k ∈ N each of these three notions agrees with theright K -Cauchyness of ( x k ). Remark 4.14.
Let ( x i : i ∈ I ) be a net in a quasi-metric space ( X, d ).1.(a) We have(4.8) i (cid:10) j ⇒ ( j i ∨ i ≁ j ) , for all i, j ∈ I . If is an order, then the reverse implication also holds for all i, j ∈ I with i = j. (b) If the net ( x i : i ∈ I ) satisfies (4.7) then it satisfies (4.4), i.e. every strong Stoltenberg-Cauchy net is Stoltenberg-Cauchy. If is an order, then these notions are equivalent.Hence, net completeness with respect to (4.4) (i.e. Stoltenberg completeness) implies netcompleteness with respect to (4.7); OMPLETENESS IN QUASI-PSEUDOMETRIC SPACES 15
2. Suppose that the net ( x i : i ∈ I ) satisfies (4.7).(a) If j, j ′ are maximal elements of I , then x j = x j ′ . Hence, if I has maximal elements, thenthere exists x ∈ X such that x j = x for every maximal element j of I , and the net ( x i ) convergesto x. (b) Consequently, the net ( x i ) also satisfies the conditions (a) and (c) from Definition 4.10,so that, GF-completeness implies completeness with respect to (4.7). Proof. i, j ∈ I with i (cid:10) j . Since j i if i, j are comparable, the implication (4.8)holds. If is an order and i = j , then j i ⇒ i (cid:10) j and i ≁ j ⇒ i (cid:10) j .(b) Since it suffices to ask that (4.4) and (4.7) hold only for distinct i, j > i ε , the equivalenceof these notions in the case when is an order follows.Suppose that the net ( x i ) satisfies (4.7). For ε > i ε ∈ I according to (4.7) and let i, j > i ε with i (cid:10) j. Taking into account (4.8) it follows d ( x i , x j ) < ε , i.e. ( x i ) satisfies (4.4).Suppose now that every net satisfying (4.4) converges and let ( x i ) be a net in X satisfying(4.7). Then it satisfies (4.4) so it converges.2.(a) Let j, j ′ be maximal elements of I . For ε > i ε according to (4.7). By Remark4.11.2, j, j ′ > i ε , j j ′ , j ′ j , so that d ( x j ′ , x j ) < ε and d ( x j , x j ′ ) < ε . Since these inequalitieshold for every ε >
0, it follows d ( x j ′ , x j ) = 0 = d ( x j , x j ′ ) and so x j = x j ′ . The convergence ofthe net ( x i ) follows from Remark 4.12.2.(c).(b) The assertions on GF-Cauchy nets follow from (a). (cid:3) We show now that completeness by nets with respect to (4.7) is equivalent to sequential right K -completeness. Proposition 4.15 ([15], Theorem 2.5) . A T quasi-metric space ( X, d ) is sequentially right K -complete if and only if every net in X satisfying (4.7) is d -convergent.Proof. We have only to prove that the sequential right K -completeness implies that every netin X satisfying (4.7) is d -convergent.Let ( x i : i ∈ I ) be a net in X satisfying (4.7). Let i k > i k − , k > , be such that (4.7) holdsfor ε = 1 / k , k ∈ N . This is possible. Indeed, take i such that (4.7) holds for ε = 1 / . If i ′ is such that (4.7)holds for ε = 2 − , then pick i ∈ I such that i > i , i ′ . Continuing by induction one obtainsthe desired sequence ( i k ) k ∈ N . We distinguish two cases.
Case I. ∃ j ∈ I, ∃ k ∈ N , ∀ k > k , i k j . Let i > j . Then for every k , i k j i implies d ( x i , x j ) < − k so that d ( x i , x j ) = 0 . Sincethe quasi-metric space (
X, d ) is T , it follows x i = x j for all i > j (see Proposition 2.1), sothat the net ( x i : i ∈ I ) is d -convergent to x j . Case II. ∀ j ∈ I, ∀ k ∈ N , ∃ k ′ > k, i k ′ (cid:10) j . The inequalities d ( x i k +1 , x i k ) < − k , k ∈ N , imply that the sequence ( x i k ) k ∈ N is right K -Cauchy (see Proposition 3.7), so it is d -convergent to some x ∈ X. For ε > k ∈ N such that 2 − k < ε and d ( x, x i k ) < ε for all k > k . Let i ∈ I, i > i k . By hypothesis, there exists k > k such that i k (cid:10) i, implying i i k ∨ i k ≁ i. Since i k i k , i , by the choice of i k , d ( x i k , x i ) < − k < ε in both of these cases. But then d ( x, x i ) d ( x, x i k ) + d ( x i k , x i ) < ε , proving the convergence of the net ( x i ) to x. (cid:3) The proof of Proposition 4.15 in the case of GF-completeness
As the result in [5] is given without proof, we shall supply one. We shall show that: A T quasi-metric space ( X, d ) is right K -sequentially complete if and only if every netsatisfying the conditions (a) and (c) from Definition 4.10 is convergent .Obviously, a proof is needed only for in the case (c).Suppose that the directed set ( I, ) has no maximal elements and let ( x i : i ∈ I ) be a net ina quasi-metric space ( X, d ) satisfying (4.4).The proof follows the ideas of the proof of Proposition 4.15 with some further details. Let i k i k +1 , k ∈ N , be a sequence of indices in I such that d ( x i , x j ) < − k for all i, j > i k with i (cid:10) j . We show that we can further suppose that i k +1 (cid:10) i k .Indeed, the fact that I has no maximal elements implies that for every i ∈ I there exists i ′ ∈ I such that(4.9) i i ′ and i ′ (cid:10) i. Let i ′ ∈ I be such that (4.4) holds for ε = 2 − . Take i such that i ′ i and i (cid:10) i ′ . Let i ′ > i be such that (4.4) holds for ε = 2 − and let i ∈ I satisfying i ′ i and i (cid:10) i ′ . Then i i and i (cid:10) i , because i i i ′ would contradict the choice of i . By induction one obtains a sequence ( i k ) in I satisfying i k i k +1 and i k +1 (cid:10) i k such that(4.4) is satisfied with ε = 2 − k for every i k .We shall again consider two cases. Case I. ∃ j ∈ I, ∃ k ∈ N , ∀ k > k , i k j . Let i > j . By (4.9) there exists i ′ ∈ I such that i i ′ and i ′ (cid:10) i, implying d ( x i ′ , x i ) < − k for all k > k , that is d ( x i ′ , x i ) = 0, so that, by T , x i ′ = x i .We also have i ′ (cid:10) j because i ′ j would imply i ′ i , in contradiction to the choice of i ′ .But then, d ( x i ′ , x j ) < − k for all k > k , so that, as above, d ( x i ′ , x i ) = 0 and x i ′ = x j .Consequently, x i = x j for every i > j , proving the convergence of the net ( x i ) to x j . Case II. ∀ j ∈ I, ∀ k ∈ N , ∃ k ′ > k, i k ′ (cid:10) j . The condition d ( x i k , x i k +1 ) < − k , k ∈ N , implies that the sequence ( x i k ) k ∈ N is right K -Cauchy, so that there exists x ∈ X with d ( x, x i k ) → k → ∞ .For ε > k ∈ N be such that 2 − k < ε and d ( x, x i k ) < ε for all k > k .Let i > i k . By II, for j = i and k = k , there exists k > k such that i k (cid:10) i . The conditions k > k , i k i, i k i k and i k (cid:10) i imply d ( x, x i k ) < ε and d ( x i k , x i ) < ε , so that d ( x, x i ) d ( x, x i k ) + d ( x i k , x i ) < ε, for all i > i k . References [1] E. Alemany and S. Romaguera,
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