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Distance Domains: Completeness
Tristan Bice
Institute of Mathematics of the Polish Academy of Sciences
Warsaw
Poland
[email protected]
Abstract.
We explore extensions of domain theoretic concepts, replacing transitive relations with general non-symmetric distances. These lead to a generalization of Smyth completeness which we characterize in various ways analogous to our previous Yoneda completeness characterizations.
Key words and phrases:
domain, distance, hemimetric, quasimetric, order, topology, complete
2010 Mathematics Subject Classification:
06A06, 18A35, 54E50, 54E55
This research has been supported by IMPAN (Poland).
Motivation
A number of works have extended domain theory – see [GHK*+*03] – from posets to more metric-like structures. However, both the classical theory and these generalizations tend to focus on just one aspect of the dual nature of domains. Our primary goal is explore the other aspect.
More precisely, the standard approach to domain theory is to start with a partial order ≤ and then define its way-below relation ≪, a transitive but generally non-reflexive relation. An alternative approach is to start with a transitive relation ≪ and then define its lower order ≤. Using maxima rather than suprema, one also obtains dual notions of completeness and continuity for ≪. This is the approach we generalize, working with a general non-symmetric distance d and its lower hemimetric d.
Also, previous works have developed quantitative domain theory in a highly category or fuzzy theoretic way – see e.g. [HW11] and [RL13]. Another goal of our paper is to provide a more classic approach through topology, metric and order theory, building on [GL13]. This leads to certain natural generalizations and should also be more accessible to analysts.
In particular, we have two examples in mind from non-commutative topology. First, consider the hereditary C*-subalgebras H(A) of a C*-algebra A, ordered by inclusion ⊆. When A is commutative, these correspond to the open subsets of a locally compact Hausdorff topological space, a well-known example of a classical domain. However, H(A) may fail to be a domain in general, even for basic non-commutative C*-algebras like C([0,1],M2)(=continuous functions from the unit interval to two by two complex matrices). The key observation here is that H(A) does, however, always form a distance domain when we replace the inclusion ordering ⊆ with the Hausdorff distance d on the positive unit balls B+1,
[TABLE]
Here the way-below distance d comes from the reverse Hausdorff distance
[TABLE]
Incidentally, (b,c)↦∥b−bc∥ is itself a natural example of a non-hemimetric distance on A+1 – see [BV18, Proposition 2.3].
There can also be merit in quantifying classical domains, e.g. consider the lower semicontinuous [0,1]-valued functions LSC(X,[0,1]) on some compact Hausdorff X with the pointwise ordering ≤. This is another well-known example of a classical domain – see [GHK*+*03, Example I-1.22]. But when we replace ≤ with
[TABLE]
we get an even nicer structure. Specifically LSC(X,[0,1]) becomes an algebraic domain, in an appropriate quantitative sense, where the finite/compact elements – see [GL13, Definition 7.4.56] – are precisely the continuous functions C(X,[0,1]) (by a slight generalization of Dini’s theorem). Moreover, this extends to the lower semicontinuous elements of A+∗∗1 for a much larger class of ordered Banach spaces A – see [Bic16].
Apart from the inherent interest in generalization, we feel examples like this justify the study of distance domains. So from now on we put functional analysis to one side to develop a general domain theory for non-symmetric distances.
Outline
While category theory is not our focus, we do consider one very elementary category GRel of generalized relations. Indeed, throughout we make use of various interpolation assumptions which are concisely described by composition ∘ in GRel. In § 1, we describe the basic properties of GRel and set out much of the notation used throughout. Note our functions take values in [0,∞], rather than the more general quantales often considered elsewhere. This is primarily to reduce the notational burden, which is already quite heavy due to the various topologies, relations and operations we need to consider. In any case, [0,∞] valued functions are perfectly suited to the analytic examples we have in mind.
As mentioned above, one of our primary goals is to generalize previous work on hemimetrics to distances, functions merely satisfying the triangle inequality. This generalization is crucial because we want to develop a dual theory of distance domains starting from distance analogs of the way-below relation. In § 2 we discuss these distances d and their associated upper and lower hemimetrics d and d.
Next, in § 3 we breifly introduce the uniform preorder ⪷ and equivalence relation ≈ on generalized relations. This generalizes the usual uniform equivalence of metrics and is needed to describe weak interpolation assumptions required for the best results (e.g. see § 5).
In § 4, we introduce balls and their associated topologies. In particular, we show how balls characterize upper and lower hemimetrics and how the preorders ≤d and ≤d defined from d coincide with the specialization preorders of ball topologies.
As we deal with non-hemimetric distances, it is natural to consider a certain strict version <d of ≤d, which we discuss in § 5. This will be particularly important in our future work when we exhibit equivalences between distance domains and classical domains of formal balls. As a preliminary to this, here we investigate the relationship between <d and ≤d under certain interpolation assumptions.
In § 6, we make some elementary observations on nets and their limits. This leads to § 7, where we discuss two natural generalizations of Cauchy nets. Note here, as elsewhere, basic properties of hemimetrics can often be extended to distances by replacing d with d and d where appropriate.
We also aim to develop the theory in a more topological way. The key here is to consider topologies generated by open holes as well as balls. In § 8 we characterize convergence in combinations of ball and hole topologies.
Yet another one of our goals is to explore the connection between topological and relational extensions of metric and order theoretic concepts.
As with hole topologies, we feel the relational notions have not received the attention they deserve. Even apart from their intrinsic interest, these relational notions can serve as a useful intermediary between classical order theoretic concepts and their topological generalizations. So in § 9 we define d-directed subsets and explore their relation to d-Cauchy nets.
Suprema are usually considered the poset analog of limits. However maxima, in an appropriate sense, can be better suited to non-reflexive transitive relations. In § 10 we extend these concepts to distances d and examine their connection to suprema and maxima relative to ≤d and <d.
In § 11, we define topological and relational notions of completeness and explain how they generalize standard notions of Yoneda, Smyth, metric and directed completeness. We then show how to turn d-Cauchy nets into d-directed subsets under several interpolation conditions. These allow d∘∙-completeness(=Smyth completeness for hemimetric d) to be derived from d-max-completeness in § 11, complementing the Yoneda completeness characterizations in [Bic18].
In our future work we will discuss generalizations of continuity and the resulting generalizations of domains, in particular showing how to complete (generalized) predomains to domains via the (reverse) Hausdorff distance and the formal ball construction.
1. Generalized Relations
The traditional category theoretic approach to quasimetric spaces is to take each quasimetric as its own category, with the elements of the space as objects and the values of the quasimetric as morphisms, as in [Law02]. Alternatively, quasimetric spaces are sometimes considered as the objects of a category with Lipschitz maps as morphisms, as in [GL13, Defintion 6.2.13]. However, the constructions we consider are best described in a category with quasimetrics, and even more general binary functions, as the morphisms instead. This is like the category of modules considered in [HW12, §2.3], except that our objects are just sets, without any distinguished hemimetric structure.
Specifically, we consider any d∈[0,∞]X×Y(= functions from X×Y to [0,∞]) as a generalized relation from X to Y. We extend the standard infix notation for classical relations to generalized relations and define
[TABLE]
Just like the category Rel of classical relations, generalized relations form the morphisms of a category GRel when composition d∘e∈[0,∞]X×Y of d∈[0,∞]X×Z and e∈[0,∞]Z×Y is defined by
[TABLE]
In fact, Rel becomes a wide subcategory of GRel when we identify each relation ⊏ ⊆X×Y with its characteristic function (as we do from now on):
[TABLE]
For any d∈[0,∞]X×Y, ⊏ ⊆[0,∞][0,∞] and r∈[0,∞] we define
[TABLE]
In particular, we let ≤d = ≤0d so
[TABLE]
Equivalently, ≤d is the relation identified with ∞d, where ∞0=0 and ∞r=∞, for r>0. Note d↦ ≤d is a left inverse of the inclusion from Rel to GRel, which is also functorial in that
[TABLE]
Various properties of Rel also extend to GRel. For example, as in [Tsa01], GRel is a category with involution dop defined by
[TABLE]
Also, GRel is a 2-category, namely a 2-poset, with the pointwise order
[TABLE]
which is compatible with both ∘ and op. Each hom-set [0,∞]X×Y is also a complete lattice with minimum 0 and maximum ∞ where, for x∈X, y∈Y and r∈[0,∞],
[TABLE]
In particular, we have ‘intersections’ d∨e and symmetrizations
[TABLE]
when X=Y, in which case we define ≡d = (≤d)∨ = ≤d∨, i.e.
[TABLE]
In fact, the only thing stopping GRel from being an allegory, in the sense of [FS90], is the modularity requirement.
However, as in division allegories, we do have Kan extensions/lifts. Namely, for d∈[0,∞]X×Z and e∈[0,∞]Y×Z, define d/e∈[0,∞]X×Y by
[TABLE]
where r+=r∨0, for r∈[0,∞], and we take ∞−∞=0. This guarantees
[TABLE]
for all a,b,c∈[0,∞]. It also means that, for all c∈[0,∞),
[TABLE]
Also, for d∈[0,∞]Z×Y and e∈[0,∞]Z×X, define e\d∈[0,∞]X×Y by
[TABLE]
Proposition \theprp.
For d∈[0,∞]X×Z, e∈[0,∞]Z×Y and f∈[0,∞]X×Y,
[TABLE]
Proof.
Simply note that, for all x∈X, y∈Y and z∈Z,
[TABLE]
2. Distances
We call d∈[0,∞]X×X a distance111Functions merely satisfying the triangle inequality do not appear to have been named before. We feel ‘distance’ is appropriate, as this is already used informally to refer to various functions which at least satisfy the triangle inequality. But if we were to follow the tradition of adding prefixes to ‘metric’ for weaker notions, ‘demimetric’ or something similar might be appropriate. if it satisfies the triangle inequality
[TABLE]
Equivalently, (2) is saying that, for all r,s∈(0,∞) and x,y,z∈X,
[TABLE]
In particular, ⊏ ⊆X×X is a distance iff it is transitive in the usual sense. As d↦ ≤d is functorial, this means ≤d is transitive whenever d is a distance. As in [GL13, Definition 6.1.1], we call a distance d a
- (1)
hemimetric if ≤d is a preorder.
2. (2)
quasimetric if ≤d is a partial order.
(Recall that a preorder is a reflexive (= ⊆ ≤) transitive relation and a partial order is an antisymmetric (≤∩≤op ⊆ =) preorder).
Non-hemimetric distances have rarely been considered until now. However, the extra generality is vital if we want to consider distance analogs of non-reflexive transitive relations, like the way-below relation from domain theory. But there are two closely related hemimetrics associated to any generalized relation, which will be crucial to our later work.
To avoid repetition, we now make the following standing assumption.
[TABLE]
Definition \thedfn.
[TABLE]
We call d and d the upper and lower hemimetric of d respectively.
This terminology is justified by the following.
Proposition \theprp.
Both d and d are hemimetrics and d=d∘d=d∘d.
Proof.
d≤(=∘d) implies d=d/d≤ = so ≤d is reflexive. As d/d≤d,
[TABLE]
Thus d is a hemimetric with d=d∘d. As dop=dop, dop and hence d is a hemimetric with dop=dop∘dop=dop∘dop and hence d=d∘d.
∎
Proposition \theprp.
If X=Y (i.e. d∈[0,∞]X×X) then222The ⇐ in (2.5) is a form of the Yoneda lemma – see [GL13, Exercise 7.5.26].
[TABLE]
Proof.
We consider d, and the d statements then follow from dop=dop.
d≤d∘d ⇔ d/d≤d.
If d≤d then d≤ =. If d≤ = then d=(d/=)≤d/d=d.
Immediate from (2.3) and (2.4).
∎
Example \thexpl.
Consider f,q∈[0,1][0,1]×[0,1] given by
[TABLE]
Here q is the restriction of the usual quasimetric on [0,∞] (note (2) for q follows from the subadditivity of +) and f is also a distance as
[TABLE]
As (x−y)+=z∈[0,1]sup(x(1−z)−y(1−z))+=z∈[0,1]sup(z(1−y)−z(1−x))+,
[TABLE]
Before moving on, we make an observation about restrictions. First, identify Z⊆Y with the characteristic function on Y×Y of = on Z, i.e.
[TABLE]
so d∘Z∘d then denotes composition restricted to Z, i.e.
[TABLE]
Proposition \theprp.
If d∘Z∘d≤d then d=d∣X×Z.
Proof.
For any w,x∈X, we see that
[TABLE]
so d∣X×Z≤d. Conversely, for any w,x∈X,
[TABLE]
where wdy≤wdz+zdy follows from d=d∘d, by § 2.
∎
3. The Uniform Preorder
As mentioned above, we usually view GRel as a 2-poset with respect to the pointwise ordering on morphisms. However, there is also a weaker 2-proset structure based on the notion of uniform equivalence for metrics. Specifically, we define the uniform preorder ⪷ by
[TABLE]
Note that that ⪷ depends only on the values of d and e close to [math]. More precisely, we show below that d⪷e is equivalent to
[TABLE]
In particular, ≈ defined by
[TABLE]
does indeed extend the usual uniform equivalence relation on metrics. Indeed, ⪷ plays a similarly fundamental role in applications (e.g. see [BV18]).
Proposition \theprp.
[TABLE]
Proof.
Assume that, for every ϵ>0, we have some δ>0 such that xey<δ implies xdy<ϵ. For any Z⊆X×Y with inf(x,y)∈Zxey=0, we have (x,y)∈Z with xey<δ so xdy<ϵ and hence inf(x,y)∈Zxdy<ϵ. Thus inf(x,y)∈Zxdy=0, as ϵ>0 was arbitrary, i.e. d⪷e.
Conversely, assume we have some ϵ>0 such that, for all δ>0, there exists some x∈X and y∈Y with xey<δ but xdy≥ϵ. In particular, we have (xn,yn) with xneyn<1/n but xndyn≥ϵ. Thus for
[TABLE]
we have inf(x,y)∈Zxey=0 but inf(x,y)∈Zxdy≥ϵ>0, i.e. d⪷e.
∎
Note for (the characteristic function of) any relation ⊏ and r∈(0,∞),
[TABLE]
Thus by § 3, ⪷ reduces to inclusion ⊆ on Rel so ⪷ is also a valid extension of the 2-poset structure from Rel to GRel.
Also note ⪷ can be expressed in terms of ed∈[0,∞][0,∞] defined by
[TABLE]
i.e. ed is the smallest monotone function satisfying
[TABLE]
Specifically, from § 3 it follows that
[TABLE]
4. Balls
Often it will also be convenient to consider the unary functions defined from binary functions by fixing one coordinate. Specifically, for x∈X and y∈Y, define xd∈[0,∞]Y and dy∈[0,∞]X by
[TABLE]
Again we identify subsets with characteristic functions so, for ⊏ ⊆X×Y,
[TABLE]
In particular, we define the open upper and lower d-balls with centre c in X or Y and radius r by
[TABLE]
These characterize d and d as follows (taking inf∅=∞).
Proposition \theprp.
[TABLE]
Proof.
If xdz<ϵ then, for any r∈(0,∞) and w∈zr∙, § 2 yields xdw≤xdz+zdw<ϵ+r so w∈xr+ϵ∙, i.e. zr∙⊆xr+ϵ∙. Conversely, say ϵ>0 and zr∙⊆xr+ϵ∙, for all r∈(0,∞), and take w∈X. If xdw=∞ then (xdw−zdw)+=0<ϵ. Otherwise, for all r∈(zdw,∞) we have w∈zr∙⊆xr+ϵ∙ and hence (xdw−zdw)+<r+ϵ−zdw. As r>zdw and w∈X were arbitrary, xdz≤ϵ. The d statement follows by duality.
∎
In particular, for any ⊏ ⊆X×Y,
[TABLE]
In [Ern91] before Lemma 3.1, these are called the ‘upper quasiorder’ and ‘lower quasiorder’ of ⊏ (we say preorder instead of quasiorder). For example, the upper and lower preorder defined from the strict ordering < on [0,∞] both coincide with the usual ordering on [0,∞], which we continue to denote by ≤ as usual. More generally, if X is a domain with way-below relation ≪ then ≪ gives back the original ordering on X. From this dual point of view, the lower preorder defined from a transitive relation is just as important as the way-below relation defined from a partial order. Our thesis is that the same is true for non-symmetric distances as well.
Let d∙ denote the topology on Y generated (as arbitrary unions of finite intersections) by open upper d-balls with centres in X, i.e.
[TABLE]
As x∞∙=⋃r∈(0,∞)xr∙ and x0∙=∅ are both d∙-open anyway, we could actually take r∈[0,∞]. Likewise, we let d∙ denote the topology on X generated by open lower d-balls with centres in Y, i.e.
[TABLE]
We discuss these further in § 8. For the moment we just note that ≤d and ≤d are the specialization preorders coming from the d∙ and d∙ topologies.
Proposition \theprp.
[TABLE]
Proof.
Note zdy=0 means xdy≤xdz, for all x∈X, which is equivalent to saying every open upper d-ball containing z must also contain y. Thus the same is true of intersections of such balls and hence unions of such intersections, i.e. all d∙-open sets. This proves (4.5) and (4.6) again follows by duality.
∎
5. The Strict Order
Here we examine a strict version <d of ≤d satisfying an analog of d=d∘d=d∘d from § 2. First, consider the following.
Proposition \theprp.
For any r,s∈(0,∞) with r<s,
[TABLE]
Proof.
If x<rdy then δ=r−xdy>0 and, for any z∈(y<δd), d=d∘d from § 2 yields
[TABLE]
Thus y∈(y<δd)⊆(x≤rd) so (x≤rd) is a d∙-neighbourhood of y.
On the other hand, if (x≤rd) is a d∙-neighbourhood of y then, in particular, x≤rdy so xdy≤r<s, i.e. x<sdy.
∎
§ 5 motivates the following definition of <d.
[TABLE]
As d is a hemimetric, x<dy is equivalent to saying there is some open upper d-ball with centre y which is entirely d-above x, i.e.
[TABLE]
When d itself is a hemimetric, § 2 yields d=d so (4.5) and (5.1) show that <d is the d∙-topological way-below relation ≺≺ familiar from Erné’s c-spaces – see [Kei17, §2.5]. In fact, if one considers the motivating example from [Kei17], namely (C0(X)+,≺≺) where f≺≺g means f≤(g−ϵ)+, for some ϵ>0, then again we see that ≺≺ is just <d for the hemimetric fdg=supx∈X(f(x)−g(x))+.
Proposition \theprp.
[TABLE]
Proof.
Note y<ϵdy, for all ϵ>0. So whenever x<dy, we have ϵ>0 with y∈(y<ϵd)⊆(x≤d), i.e. x≤dy and hence ≤d ⊇ <d.
If x≤dy<dz then, for some ϵ>0, (z<ϵd)⊆(y≤d)⊆(x≤d), as d=d∘d, by § 2, so ydw=0 implies xdw≤xdy+ydw=0. Thus x<dz and hence ≤d∘<d ⊆ <d.
If x<dy≤dz then, for some ϵ>0, (z<ϵd)⊆(y<ϵd)⊆(x≤d), as d is a distance, by § 2, so zdw<ϵ implies ydw≤ydz+zdw<ϵ. Thus x<dz and hence <d∘≤d ⊆ <d.
The reverse inclusions are immediate from the fact ≤d and ≤d are reflexive, by § 2.
If x≤dy<dz then, for some ϵ>0, (z<ϵd)⊆(y≤d)⊆(x≤d), as d=d∘d, by § 2, so ydw=0 implies xdw≤xdy+ydw=0. Thus x<dz and hence ≤d∘<d ⊆ <d.
∎
Corollary \thecor.
If d∈[0,∞]X×X is a distance, <d is transitive and
[TABLE]
Proof.
As d is a distance, d,d≤d, by § 2, so ≤d,≤d ⊇ ≤d and <d ⊆ <d. Thus, by § 5,
[TABLE]
By § 5, <d = <d∘≤d and hence, by § 1,
[TABLE]
We can improve this to an equality under a certain interpolation condition.
Proposition \theprp.
If \ \overline{\mathbf{d}}\ \circ<^{\mathbf{d}}\ \leq\mathbf{d}\ then ≤d =<d.
Proof.
To prove <d⊆ ≤d, say z≤dy, i.e. zdy>0, so we have x∈X with xdy−xdz>0, i.e. xdz<xdy. As d ∘<d ≤d, we have w∈X with xdw<xdy and w<dz. Thus 0<xdy−xdw≤wdy, as d=d∘d by § 2, i.e. w≰dy and hence w<dy, so z<dy.
∎
The similar condition d∘<d≤d can be derived from another interpolation condition involving dP. Specifically, for d∈[0,∞]X×Y, define dP on X×P(Y), where P(Y)={Z:Z⊆Y}, by
[TABLE]
In particular, note x≤dPZ means x≤dz, for all z∈Z. Also consider the following condition on closed upper balls xr∙={y∈Y:xdy≤r} with finite radius r<∞.
[TABLE]
Proposition \theprp.
[TABLE]
Proof.
Take any x∈X and Z⊆Y and let r=x(dP)Z=supz∈Zxdz. If r=∞ then we immediately have x(d∘≤dP)Y≤r. Otherwise, we have a ≤d -minimum y of xr∙. Thus xdy≤r and y≤dz, for all z∈Z, i.e. y≤dPZ. So x(d∘≤dP)Z≤r=x(dP)Z, proving the first ⇒.
For the second ⇒, assume d∘≤dP≤dP and say xdy<r. Take δ with 0<δ<r−xdy so that, as d=d∘d, by § 2,
[TABLE]
So we have z∈Y with xdz<r and z≤dP(y<δd) and hence z<dy. Thus x(d ∘<d)y<r. As r>xdy was arbitrary, d∘<d≤d.
∎
When d is a hemimetric, we can even weaken ≤ to ⪷.
Proposition \theprp.
If d is a hemimetric then
[TABLE]
Proof.
Assume d∘≤dP⪷dP and say xdy<r. By § 3, we have some δ>0 such that y(dP)Z≤δ implies y(d∘≤dP)Z<r−xdy. In particular, we can take Z=yδ∙ and then we have z∈X with ydz<r−xdy and z≤dPyδ∙. Thus xdz≤xdy+ydz<r and (y<δd)=(y<δd)⊆(z≤d), i.e. z<dy. As r>xdy was arbitrary, d∘<d≤d.
∎
Corollary \thecor.
If \ \overline{\mathbf{d}}\ \circ\leq^{\mathbf{d}}\ \leq\,\mathbf{d}\ and \ \underline{\mathbf{d}}\circ\mathbin{\leq^{\underline{\mathbf{d}}\mathcal{P}}}\,\precapprox\,\underline{\mathbf{d}}\mathcal{P}\ then ≤d =<d.
Proof.
By § 5, § 2 and § 5 (for d),
[TABLE]
Thus ≤d =<d, by § 5.
∎
For example, (5) and hence ≤d =<d holds in C0(X)+, where again fdg=supx∈X(f(x)−g(x))+. Indeed, for any f∈C0(X)+ and r∈[0,∞], we see that (f−r)+ is the ≤d-minimum of the closed upper d-ball fr∙ with centre f and radius r.
But if we consider the opposite hemimetric on C0(X) given by feg=supx∈X(g(x)−f(x))+ and X is not compact then <e is vacuous, owing the fact any f,g∈C0(X)+ must vanish at infinity. This means <e is trivial, i.e. f<eg for arbitrary f,g∈C0(X)+. On the other hand, here ≤e = ≤e is just the opposite of the pointwise ordering on C0(X)+. In particular, ≤e is not trivial, so the inclusion in (5.5) is strict.
Also d ∘<d ≤d and hence ≤d =<d holds in spaces formal balls, which will be crucial in our future work when we look at generalized (pre)domains.
6. Nets
We consider nets in a slightly more general sense than usual. Specifically, as we deal with non-hemimetric distances, we must also deal with non-reflexive nets (to allow for d-Cauchy nets even when ≤d is not reflexive). So by a net we mean a non-empty set indexed by a directed set Λ, i.e. we have (possibly non-reflexive) transitive ≺⊆Λ×Λ satisfying
[TABLE]
As usual, we define limits by
[TABLE]
In fact, these are also the limits with respect to the preorder ⪯ given by
[TABLE]
as in (4.4). Also note that is suffices to verify (6.1) for all open O in a subbasis S for the topology. Indeed, as nets are indexed by directed sets, if (6.1) holds for all O∈S then (6.1) holds for all finite intersections of elements of S and hence for all unions of finite intersections of elements of S, i.e. all open sets. In particular, for any topologies T and U, convergence in their supremum T∨U(= the topology with subbasis T∪U) is the same as convergence in both T and U, i.e.
[TABLE]
Limits in [−∞,∞] are considered with respect to the usual interval topology and limits inferior and superior are defined as usual by
[TABLE]
Note limits inferior/superior are below/above infima/suprema, i.e.
[TABLE]
Also, (rλ) converges in [−∞,∞] iff
[TABLE]
in which case limλrλ=limsupλrλ=liminfλrλ.
We also use a number of standard facts like
[TABLE]
Note these are only valid when we do not end up with ∞−∞ in the middle, which is not a problem on [−t,∞], for any t∈[0,∞). Also,
[TABLE]
Indeed, in the finite case this follows from (6.6) by taking sλ=−rλ, while the infinite case can be verified directly. Also, as r↦r+ is continuous and (non-strictly) increasing on [−∞,∞], we have
[TABLE]
For example, combining these facts yields
[TABLE]
as long as s or liminfλrλ is finite.
Let us adopt the convention that when nets are written on the left of d we take the limit superior, while on the right we take the limit inferior:
[TABLE]
We also extend this notation to unary functions, defining
[TABLE]
(The limits here are pointwise, i.e. in the product topology of [0,∞]X).
To avoid repetition, from now on we assume X=Y, i.e.
[TABLE]
Proposition \theprp.
For any (zλ)⊆X,
[TABLE]
Proof.
By (6.6),
[TABLE]
∎
7. Cauchy Nets
Definition \thedfn.
For any net (xλ)⊆X, define
[TABLE]
Equivalently, (xλ) is d-Cauchy if and only if
[TABLE]
when we consider ≺ itself as a directed subset of Λ×Λ with respect to the product ordering ≺×≺. These nets are ‘increasing modulo ϵ’, in a certain sense. More precisely, they can be characterized by <ϵd:
[TABLE]
In particular, if ⊏ is a transitive relation then the ⊏-Cauchy nets are precisely the increasing nets, at least beyond a certain point γ0. On the other hand, the ⊏-pre-Cauchy nets are the ‘directed nets’ from [GHK*+*03, Definition O-1.2]. In the literature on hemimetrics, d-Cauchy nets are more often considered than d-pre-Cauchy nets (a notable exception is [Wag97], where sequences that we would call pre-Cauchy/Cauchy are called Cauchy/strongly Cauchy respectively). However, most results on d-Cauchy nets can be generalized to d-pre-Cauchy nets without difficulty, as we demonstrate, and these are sometimes more convenient to work with (e.g. it suffices to consider d-pre-Cauchy nets indexed by posets, while with d-Cauchy nets we must consider more general transitive relations).
On the other hand, from a metric space point of view, both (7.1) and (7.2) extend the usual notion of a Cauchy net.
Proposition \theprp.
If d is a symmetric distance, i.e. d=dop≤d∘d,
[TABLE]
Proof.
The ⇒ part is immediate. Conversely, if (xλ)⊆X is d-pre-Cauchy then, for every ϵ>0, we have α,β such that, for all γ≻β, xαdxγ<ϵ. Thus, for all δ≻γ, d=dop≤d∘d yields xγdxδ≤xαdxγ+xαdxδ<2ϵ, i.e. (xλ) is d-Cauchy.
∎
Here are a few basic but important facts about pre-Cauchy nets. Note a version of (2) below appears in [Wag97, Theorem 2.26].
Theorem 7.1**.**
- (1)
If (xλ)⊆X is d-pre-Cauchy then (xλ) has a d-Cauchy subnet.
2. (2)
If (xλ)⊆X is d-pre-Cauchy then xλd converges (pointwise).
3. (3)
If (xλ)⊆X is d-pre-Cauchy then dxλ converges (pointwise) and
[TABLE]
4. (4)
If (xλ)⊆X is d-pre-Cauchy and d is a distance then
[TABLE]
Proof.
If Λ is finite then it has a maximum γ, which means the single element net xγ is a d-Cauchy subnet. Otherwise, let ∣F∣ denote the cardinality of F and consider the finite subsets of Λ
[TABLE]
directed by ⫋. We define a map f:F(Λ)∖{∅}→Λ recursively as follows. Let f({λ})=λ, for all λ∈Λ. Given F∈F(Λ)∖{∅}, take f(F)∈Λ such that, for all E⫋F, f(E)≺f(F) and
[TABLE]
In particular, λ≺f(F) whenever λ∈F={λ}. This means that {f(F):F∈F(Λ)∖{∅}} is cofinal in Λ and hence (xf(F)) is a subnet of (xλ), which yields the second ≤ in
[TABLE]
Thus (xf(F)) is a d-Cauchy subnet of (xλ).
If (xλ) is d-pre-Cauchy then, for all y∈X,
[TABLE]
Thus xλdy converges, by (6.4).
If (xλ) is d-pre-Cauchy then, for all y∈X,
[TABLE]
Thus ydxλ converges, by (6.4).
First note that
[TABLE]
For the converse, take ϵ∈(0,∞) and replace the d-pre-Cauchy net (xλ) with a subnet if necessary so that, for all γ,
[TABLE]
Note this suffices to prove the result for the original net as we already know that xdxλ converges, by (3), and xλdy converges, by (2) (note applying d to the hemimetric d leaves it unchanged)
We first claim that, for all z∈Z,
[TABLE]
If liminfγzdxγ<∞ then this follows from (6.8). If liminfγzdxγ=∞ then, using the fact that d=d∘d by § 2,
[TABLE]
for all γ. Thus ∞=zdxγ, for all γ, so
[TABLE]
again proving the claim.
Now consider
[TABLE]
As zdxγ≤zdxλ+xλdxγ, by d=d∘d from § 2, it follows that −zdxλ≤−zdxγ+xλdxγ, by (1.1), and hence
[TABLE]
By (2.3), d≤d so we have d(xλ)≤d(xλ). Conversely,
[TABLE]
Again by (2.3), d≤d so (xλ)d≤(xλ)d, while conversely,
[TABLE]
8. Holes
Define the open upper/lower holes with centre c∈X and radius r by
[TABLE]
Note these are defined just like open balls in (4.1) and (4.2) but with < reversed. Let d∘, d∘, d∘∘, d∙∙, d∘∙ and d∙∘ denote the topologies generated by the corresponding balls and holes, i.e. by arbitrary unions of finite intersections, e.g. d∘ is the topology with subbasis (x∘r)x∈X,r∈(0,∞) and d∘∙=d∙∨d∘ etc.. As with balls, we could even take r∈[0,∞], as x∘0=⋃r∈(0,∞)x∘r and ∅=x∘∞. Beware that in general these subbases are not bases – for hemimetric d, the balls form a basis for the ball topologies, by [GL13, Lemma 6.1.5], but even this can fail for more general distances.
Up until now, most of the literature has focused on ball topologies. However, as mentioned in [GL13, Exercise 6.2.11], hole topologies generalize the upper topology from order theory. This allows for simple generalizations of certain order theoretic concepts. Also, the double hole topology d∘∘ coincides with various kinds of weak topologies, although this too does not appear to be widely recognized. For example, the double hole topology is the usual product topology on products of bounded intervals, the weak operator topology on projections on a Hilbert space and the Wijsman topology on subsets of X (see [Bic15, Examples 5 and 6 and §5.3]).
We denote convergence in d∙, d∘, d∘∙, etc. by →∙, →∘, →∘∙, etc..
Proposition \theprp.
For any net (xλ)⊆X,
[TABLE]
Proof.
Recall that for convergence it suffices to consider subbasic open sets, in this case the balls y∙r, for y∈X and r∈(0,∞). So xλ→∙x means that, for all y∈X and r∈(0,∞), if x∈y∙r then (xλ)λ≻γ⊆y∙r, for some γ. Thus if xdy<r then limsupλxλdy≤r. As r and y were arbitrary, this means limsupλxλdy≤xdy and hence (xλ)d≤xd. Conversely, if (xλ)d≤xd, i.e. limsupλxλdy≤xdy, for all y∈X, then xdy<r implies that limsupλxλdy<r, for all r∈(0,∞), and hence xλ→∙x.
Likewise xλ→∘x means that, for all y∈X and r∈(0,∞), if x∈y∘r then (xλ)λ≻γ⊆y∘r, for some γ. Thus if ydx>r then liminfλydxλ≥r. As r and y were arbitrary, this means liminfλydxλ≥ydx and hence d(xλ)≥dx. Conversely, if d(xλ)≥dx, i.e. liminfλydxλ≥ydx, for all y∈X, then ydx>r implies that liminfλydxλ>r, for all r∈(0,∞), and hence xλ→∘x.∎
Likewise,
[TABLE]
As xλ→∘∙x if and only if xλ→∙x and xλ→∘x, by (6.2), and rλ→r if and only if limsupλrλ≤r≤liminfλrλ, and likewise for xλ→∙∘x, we have
[TABLE]
In general, these convergence notions depend on all d values, not just the small ones. In particular, without extra assumptions, they can not be characterized by statements like xλdx→0 familiar from metric space theory. However, there are still some general relationships of this sort.
Proposition \theprp.
[TABLE]
Proof.
Recall from § 2 that d=d∘d=d∘d.
If xλdx→0 then cdx≤liminfλ(cdxλ+xλdx)=cd(xλ).
If xλdx→0 then, as d=d∘d yields xλdx+xdc≥xλdc, (1.1) yields xdc≥limsupλ(xλdc−xλdx)=(xλ)dc.
If xλ→∙x≤dx then limsupλxλdx=(xλ)dx≤xdx=0.
If xλd∨x→0 then xλdx→0 so xλ→∘x, by (8.7), but also xλdopx=xλdopx→0 so xλ→∙x, by (8.8).
∎
In [GL13] Definition 7.1.15, any x with (xλ)d=xd is called a d-limit of (xλ) (these are called forward limits in [Bonsangue1998] before Proposition 3.3 and just limits in [KS02] Definition 11). In general, d-limits are not true limits in any topological sense, as they are not preserved by taking subnets. For example, if we consider xdy=(x−y)+ on {0,1} and take the sequence (xn) defined by x2n=0 and x2n+1=1, for all n, then (xn)d=1d while (x2n)d=0d. But for d-pre-Cauchy nets, d-limits are d∙∘-limits.
Proposition \theprp.
If (xλ) is d-pre-Cauchy with subnet (yγ) then
[TABLE]
Proof.
If xλ→∙∘x, i.e. limλxλd=xd (see (8.6)) then certainly limsupλxλd=xd, i.e. (xλ)d=xd. Conversely, if limsupλxλd=xd then limλxλd=xd, as xλd converges, by Theorem 7.1 (2). Likewise, as xλd converges, limλxλd=limyγd, for any subnet (yγ), so limλxλd=xd if and only if limγyγd=xd.
Apply Theorem 7.1 (3) as above.
By (7.3) and (8.12),
[TABLE]
As (xλ)dx=xdx=0, (8.7) yields xλ→∘x. On the other hand, xdy=(xλ)dy=ydy=0, where the second equality follows (8.13) and the xλ→∘∙y assumption. As d≤d∘d, it follows that dy≤dx+xdy=dx. Then (xλ)dop=d(xλ)=dy≤dx=xdop, i.e. xλ→∙x, where the first and second equalities follow from (8.5).
By (7.4), (xλ)d=(xλ)d. Thus it suffices to prove
[TABLE]
for d-pre-Cauchy (xλ). The ⇐ part is (8.7). Conversely, (7.3) yields
[TABLE]
where the inequality follows from xλ→∘x and (8.2).
If xλ→∘∘x then xλdx→0, by (8.15), so xdx≤(xλ)dx=0, i.e. x≤dx. Also xλdx≤xλdx→0, as d≤d by (2.3), so xλ→∙x, by (8.8). This proves ⇒, while (8.9) and (8.15) prove ⇐.∎
For hemimetric d, (8.8) and (8.9) show that d∙-convergence is equivalent to the statement xλdx→0 familiar from metric space theory. But for general distance d, it is rather d∘-convergence that is characterized by xλdx→0, at least for d-pre-Cauchy nets, by (8.15).
Also note (8.13) and (8.14) describe a close relationship between d∘∙-limits and d∘∘-limits of d-pre-Cauchy (xλ) (as (8.11) and (8.16) show x=d∘∘-limxλ iff (xλ)d=xd). Namely, every d∘∙-limit of a d-pre-Cauchy net (xλ) is a d∘∘-limit, by (8.13), while conversely the mere existence of a d∘∙-limit guarantees that any d∘∘-limit is a d∘∙-limit, by (8.14).
For a simple example of a d-Cauchy net where xλ→∙∘x≰dx and hence xλ→∘∘x, take any xλ→0<x in [0,∞], taking ydz=z for d.
9. Directed Subsets
Directed subsets play a fundamental role in domain theory. These correspond to increasing nets which are generalized by the (pre-)Cauchy-nets above, and this is usually considered the only path to quantitative domain theory. However, an equally valid but subtly different theory can be obtained from a more direct generalization of directed subsets.
Definition \thedfn.
We call Y⊆X d-directed if, for all finite F⊆Y,
[TABLE]
Equivalently, Y is d-directed if and only if
[TABLE]
where F(Y) again denotes the finite subsets of Y. In particular, for any transitive relation ⊏, Y is ⊏-directed iff every finite subset of Y has an upper bound w.r.t. ⊏, i.e. iff Y is directed in the usual sense.
It will also be convenient to consider the following weaker notion obtained by restricting to singleton F.
Definition \thedfn.
We call Y⊆X d-final if, for all x∈Y,
[TABLE]
Equivalently, Y is d-final if and only if
[TABLE]
In particular, for any transitive relation ⊏, Y is ⊏-final iff every single element x has an upper bound y⊐x. In [Kei17], ⊏-final subsets are called ‘cofinal’, while in [GHK*+*03, Proposition III-4.3] and [GL13, Proposition 5.13] they are called ‘rounded’, at least in the ideal case. Note arbitrary subsets are d-final when ≤d is reflexive. In particular, arbitrary subsets are ⊑-final when ⊑ is a preorder.
As with nets, let us adopt the convention that sets written on the left/right of a function denote suprema/infima, so
[TABLE]
Again we extend this to unary functions, i.e.
[TABLE]
For example, applying these conventions twice, for any Y,Z⊆X we have
[TABLE]
So the definition of d-directedness can thus be restated as follows
[TABLE]
In fact, for d-directed Y, it does not matter where we put the parentheses.
Proposition \theprp.
If d is a distance and Y is d-final then
[TABLE]
Proof.
If Y is d-final and, for all F∈F(X), (Fd)Y=F(dY) then in particular, for all F∈F(Y), we have (Fd)Y=F(dY)=0, i.e. Y is d-directed.
For each x∈F, xdY≤(Fd)Y so F(dY)≤(Fd)Y. Conversely, say Y is d-directed and take ϵ>0. For each x∈F, we have x′∈Y with xdx′≤xdY+ϵ≤F(dY)+ϵ. Then we can take y∈Y with F′dy<ϵ, where F′={x′:x∈F}. If d is a distance then Fdy≤F(dY)+2ϵ. As ϵ>0 was arbitrary, (Fd)Y≤F(dY).
If d is a distance then Yd≤Yd, by (2.3). Conversely, note first that r∈R,s∈Sinf(r+s)≤infR+supS, for all R,S⊆[0,∞], so
[TABLE]
So if Y is also d-final then
[TABLE]
Likewise dY≤dY, by (2.3), and conversely
[TABLE]
Recall the standard topological notion of separability, namely that X is T-separable, for some topology T on X, if X contains a countable T-dense subset Y, i.e. if every non-empty O∈T contains some y∈Y.
Proposition \theprp.
If d is a distance then
[TABLE]
Proof.
Assume Z is d∙-dense in X. If X is d-final then, for all x∈X and ϵ>0, xϵ∙ is non-empty and hence contains some z∈Z, i.e. X(dZ)=0. If X is d∙-separable then we can choose Z to be countable, proving ⇒.
Conversely, if X(dZ)=0 then certainly X(dX)=0, i.e. X is d-final. And if O=(x1)ϵ1∙∩⋯∩(xn)ϵn∙ is non-empty, for some x1,⋯,xn∈X and ϵ1,⋯,ϵn>0, then we can take x∈O and ϵ>0 such that xkdx+ϵ<ϵk, for all k≤n. As d is a distance, this means xϵ∙⊆O. As X(dZ)=0, we have some z∈Z with z∈xϵ∙⊆O, so Z is indeed dense in X.
∎
It will be useful to define what it means for a subset to be below a net and vice versa. Specifically, for any (xλ)⊆X and Y⊆X, let
[TABLE]
Proposition \theprp.
For any (xλ)⊆X and Y⊆X,
[TABLE]
Proof.
As d=d∘d=d∘d, by § 2, Y≤d(xλ) yields
[TABLE]
Again as d=d∘d=d∘d, by § 2, Y≥d(xλ) yields
[TABLE]
Note that if (yγ) is a subnet of (xλ) then
[TABLE]
The converses also hold for pre-Cauchy nets.
Proposition \theprp.
If (yγ) is a subnet of (xλ) then
[TABLE]
Proof.
Assume (xλ) is d-pre-Cauchy and (yγ)≤dY. Then
[TABLE]
Thus (xλ)≤dY.
If (xλ) is d-pre-Cauchy then ydxλ has a limit, for any y, by Theorem 7.1 (3). So if ydyγ=0, for some subnet (yγ), this limit must be [math]. Applied to all y∈Y, we see that Y≤d(yγ) implies Y≤d(xλ).∎
Defining Y≤dx to mean y≤dx, for all y∈Y, we also see that
[TABLE]
Indeed if y∈Y≤d(xλ) and xλ→∘x then ydx≤yd(xλ)=0, by (8.2).
Different versions of quantitative domain theoretic concepts are connected via results about d-directed subsets having equivalent d-pre-Cauchy nets (and vice versa, a topic we will return to in § 11). Specifically, let
[TABLE]
Proposition \theprp.
For any Y⊆X,
[TABLE]
Proof.
If Y is d-directed then, for F∈F(Y) and ϵ>0, take yF,ϵ∈Y with FdyF,ϵ<ϵ. Ordering F(Y)×(0,∞) by ⊆×≥, we get (yF,ϵ)⊆Y≤d(yF,ϵ). In particular, (yF,ϵ) is d-pre-Cauchy. By Theorem 7.1 (1), we can replace (yF,ϵ) with a d-Cauchy subnet. Lastly, note (yF,ϵ)⊆Y implies (yF,ϵ)(dY)≤Y(dY)=0, as Y is d-directed and hence d-final, i.e. (yF,ϵ)≤dY.
If (xλ)≡dY then, as d is a distance,
[TABLE]
If (xλ)≡dY then, for any F∈F(Y),
[TABLE]
as (Fd)(xλ)=0 because Y≤d(xλ) and (xλ)(dY)=0 because (xλ)≤dY. This shows Y is d-directed. The converse is (9.8).
Assume d is a distance, X is d∙∙-separable and Y is d-directed. As d is hemimetric, d∙∙=d∨∙, by [GL13, Proposition 6.1.19]. Also X is trivially d∨-final, so we have countable Z⊆X with X(d∨Z)=0, by § 9. Let (zn)n∈N enumerate Z (note we do not consider [math] to be an element of N). For each n∈N, we can take y1,⋯,yn∈Y with zkdyk<zkdY+1/n, for all k≤n. Applying § 9 to F={y1,⋯,yn}, we obtain xn∈Y with Fdxn<1/n. As d is a distance, this implies that zkdxn<zkdY+2/n, for all k≤n. For any y∈Y and ϵ>0, we have N∈N with y(d∨)zN<ϵ and hence zNdY≤zNdy+ydY<ϵ, as Y is d-final. Thus, for any n≥N,
[TABLE]
As ϵ>0 was arbitrary, ydxn→0, so (xn)⊆Y≤d(xn). This completes the proof of ⇐, while ⇒ follows from (9.10).∎
Mostly we use d-directed subsets, but they can be replaced by d-ideals.
Definition \thedfn.
We call I⊆X a d-ideal if, for all F∈F(X),
[TABLE]
Note that for the ⇐ part it suffices to consider singleton F, i.e.
[TABLE]
For if (Fd)I=0 then certainly xdI=0, for all x∈F, so (9.12) yields x∈I, for all x∈F, and hence F⊆I.
Proposition \theprp.
For distance d, the d∙-closure of d-final Y⊆X is
[TABLE]
If Y is d-directed then Y∙ is the smallest d-ideal containing Y.
Proof.
Assume d is a distance and xdY=0. Then whenever we have c1,⋯,cn∈X and r1,⋯,rn∈(0,∞) with x∈(c1)r1∙∩…∩(cn)rn∙, we can always find y∈Y with xdy<(r1−c1dx)∧…∧(r1−c1dx), as xdY=0. It follows that y∈(c1)r1∙∩…∩(cn)rn∙, as d is a distance. Thus x∈Y∙(= the d∙-closure of Y). Conversely, if ≤d is reflexive and xdY>ϵ>0 then xϵ∙∩Y=∅ while x∈xϵ∙, i.e. x∈/Y∙. Thus if d is a hemimetric,
[TABLE]
If d is a distance and Y is d-final then (9.2) and the above argument applied to the hemimetric d shows the d-closure Y∙ is given by (9.13):
[TABLE]
It follows that any d-ideal I containing Y contains Y∙, for if 0=xdY≥xdI then x∈I, by (9.12). But if Y is d-directed then, by (9.1),
[TABLE]
For the last ⇔, note that (Fd)Y∙≤(Fd)Y, as Y⊆Y∙, and conversely
[TABLE]
by (9.13). Thus Y∙ itself is a d-ideal.
∎
Proposition \theprp.
If I is d-ideal then I is d-directed and d∙-closed. If d is a distance, any d-directed d∙-closed I⊆X is a d-ideal.
Proof.
If I is d-ideal then certainly I is d-directed. In particular, I is d-final so dI≤dI follows as in the proof of (9.2):
[TABLE]
So if x is in the d∙-closure of I then xdI≤xdI=0, by (9.13) (with d replacing d). Thus x∈I, by the definition of d-ideal, i.e. I is d∙-closed.
Conversely, assume d is a distance and I is d-directed and d∙-closed. In particular, the ⇒ part of § 9 holds, as I is d-directed. Also any x with xdI=0 is in I, by (9.13), as d is a distance and I is d∙-closed and d-final (even d-directed). This implies that the ⇐ part of § 9 holds too, as noted in (9.12).
∎
10. Upper Bounds
Next we examine ‘d-minimal’ upper bounds of d-directed subsets.
Definition \thedfn.
Define d-suprema and d-maxima of Y⊆X by
[TABLE]
Note d-suprema and d-maxima are not necessarily unique, so = here is not really equality. Put another way, we are officially taking d-sup and d-max as relations, not functions, and adding the = symbol simply for consistency with standard supremum/maximum notation. We consider d-suprema and d-maxima analogous to d∘∘-limits and d∘∙-limits respectively, as indicated by the following analog of § 8.
Proposition \theprp.
If d is a distance then, for any Y⊆X,
[TABLE]
Proof.
If Yd=xd and x≤dx then Ydx=xdx=0, i.e. Y≤dx so x=d-supY. If Y≤dx and xd≤Yd then xdx≤Ydx=0 and, as d is a distance, Yd≤Ydx+xd=xd, i.e. x≤dx and xd=Yd.
If dY≤dx then Yd≥xd as
[TABLE]
Also d≤d, as d is a distance, so Y≤dx implies Y≤dx.
If dY=dx then, as Y is d-final, 0=ydY=ydx, for all y∈Y, i.e. Y≤dx so x=d-maxY. Conversely, as d is a distance, Y≤dx implies dx≤dY+Ydx=dY.
If x=d-supY and y=d-maxY then xdy≤Ydy≤Ydy=0, as d is a distance. So dY=dy≤dx+xdy=dx. As Y is d-final and Y≤dx, Ydx≤Y(dY)+Ydx=0, i.e. Y≤dx too so x=d-maxY.∎
For any ⊏ ⊆X×X, we see that
[TABLE]
Thus if ⪯ is a partial order then ⪯-suprema and ⪯-maxima are suprema and maxima in the usual sense with respect to ⪯. Indeed, if ⊏ is antisymmetric and x⊏x= ⊏-maxY then, for some y∈Y, we have x⊏y⊏x and hence x=y. Maxima are more interesting for non-reflexive relations, like the way-below relation ≪ from domain theory or even just the strict ordering < on R. Then maxima can be intuitively more like suprema, e.g. for any Y⊆R,
[TABLE]
We can also relate d-suprema and d-maxima to ≤d-suprema and <d-maxima, at least under certain interpolations assumptions. One of these involves Pd∈[0,∞]P(X)×X (not to be confused with dP) defined by
[TABLE]
So Y≤Pdx means Ydx=0, i.e. Y≤dx.
Proposition \theprp.
For any Y⊆X,
[TABLE]
Proof.
Multiplying xd≤Yd by ∞ yields (x≤d)⊇(Y≤d).
Assume x=≤d-supY=d-supY so Ydz<xdz, for some z∈X. As (≤Pd∘d)≤Pd, we have w∈X such that wdz<xdz and Y≤dw and hence x≤dw. Then xdz≤xdw+wdz<xdz, a contradiction.
Assume x=<d-maxY=d-maxY so zdx<zdY, for some z∈X. As (d∘<d)≤d, we have w<dx with zdw<zdY. This means that wdY≥zdY−zdw>0 so, for all y∈Y, w≰dy and hence w<dy, contradicting x=<d-maxY.
Assume x=d-maxY. As Y is <d-final, for any y∈Y, we have z∈Y with y<dz≤dx and hence y<dx, by (5.3), i.e. Y<dx. Now take z∈X with z<dx. We need to show that z<dy, for some y∈Y. As <d∘≤d⊇<d, we can take w∈X with z<dw≤dx, so (w<ϵd)⊆(z≤d), for some ϵ>0. As w≤dx=d-maxY, we have y∈Y such that wdy≤wdy<ϵ and hence z≤dy. As Y is <d-final, we have y′∈Y with y<dy′ so z<dy′, by § 5. ∎
11. Completeness
Next we consider generalizations of metric and directed completeness.
Definition \thedfn.
For any topology T on X and relation R⊆X×P(X),
[TABLE]
When d is clear, we simply refer to T-completeness and R-completeness. The cases of primary interest are T=d∘∘, d∘∙ and R=d-sup, d-max.
When d is a distance and T=d∘∙ or d∘∘, we can replace d-Cauchy with d-pre-Cauchy, by Theorem 7.1 (1) and § 8. In the hemimetric case, these are usually called Smyth and Yoneda completeness – see [GL13, Definitions 7.2.1 and 7.4.1] – as § 8 and (8.15) then show that d∙-limits and d∘-limits of d-Cauchy (xλ) coincide.
If d is a metric then these are all equivalent to the usual notion of metric completeness – see [GL13, Lemma 7.4.3].
On the other hand, for any poset (X,⊑)
[TABLE]
(where ⊑∘∘ is topology generated by ⊑-holes (x⊑) and (⊑x)).
If ⊑ is the lower preorder of some transitive ⊏ on X then, moreover,
[TABLE]
(where ⊏∘∙ is topology generated by upper ⊏-balls (x⊏) and lower ⊏-holes (x⊏)). However if d is a metric, every d-directed subset contains at most 1 element, making X trivially d-sup-complete and d-max-complete. So unlike the topological notions of completeness, the relational notions do not generalize metric completeness. Indeed, the topological notions are stronger (even for non-distance d), as we now show.
Proposition \theprp.
[TABLE]
Proof.
Take d-directed Y⊆X, so we have d-Cauchy (xλ)≡dY, by (9.8). Note we can take (xλ)⊆Y by taking Y as the ambient space X in (9.8).
If X is d∘∘-complete, we have x∈X with xλ→∘∘x. As xλ→∘x, (9.7) yields Y≤dx. As xλ→∘x, (8.4) and (xλ)⊆Y yield
[TABLE]
If X is d∘∙-complete, we have x∈X with xλ→∘∙x. As xλ→∘x, (9.7) yields Y≤dx. As xλ→∙x, (8.3) and (xλ)⊆Y yield
[TABLE]
Conversely, we can derive the topological from the relational notions under various interpolation conditions (whose naturality/applicability will be indicated by some closely related conditions as well as examples like C0(X)+). This was done for d∘∘ and d-sup in [Bic18] and here we aim to do the same for d∘∙ and d-max. First we use these conditions to turn d-pre-Cauchy nets into equivalent subsets and sequences, collecting their corollaries for completeness at the end.
Unlike much of the rest of the paper, these results have no real analogs in either metric or order theory. Indeed, if d is a transitive relation ⊏ then ⊏∘∙-completeness and ⊏-max-completeness are automatically equivalent. In this case, any ⊏-pre-Cauchy net can be turned into an equivalent ⊏-directed subset by using Theorem 7.1 (1) to obtain a ⊏-increasing subnet (which becomes a ⊏-directed subset when we forget the indexing set). On the other hand, as mentioned above, d-max-completeness holds trivially for any metric d and will thus be no help at all in verifying d∘∙-completeness, i.e. metric completeness. Consequently, the results below will become either trivial or inapplicable in these classical cases.
Our first result is a converse of (9.9) based on [Bic18, Theorem 1]. It relies on the interpolation condition d∘≤dP⪷dP which, in the hemimetric case, weakens the middle condition considered in § 5. This condition applies to spaces of formal balls, as we discuss in our future work, and the space C0(X)+, where again fdg=supx∈X(f(x)−g(x))+. Indeed, (5) applies to C0(X)+, by the comments after § 5, so d∘≤dP⪷dP also applies, by § 5. However, note that d∘≤dP⪷dP does not apply to any metric space with at least two points.
Theorem 11.1**.**
If d is a distance and d∘≤dP⪷dP then
[TABLE]
Proof.
As r→0limdPd∘≤dP(r)=0, we can define rn↓0, i.e. a strictly decreasing sequence (rn) with rn→0, such that
[TABLE]
Take d-pre-Cauchy (xλ)⊆X. If necessary, we can replace (xλ) with a d-Cauchy subnet, by Theorem 7.1 (1), and the conclusion of the theorem will be preserved, by § 9 (noting that, as d is a distance, any d-pre-Cauchy net is both d-pre-Cauchy and d-pre-Cauchy, by (2.3)). Define f:F(Λ)∖{∅}→Λ (where F(Λ) denotes the finite subsets of Λ) recursively as follows. Let f({λ})=λ and, given F∈F(Λ) with ∣F∣>1, take f(F)≻f(E), for all E⫋F, such that
[TABLE]
Now xf(F)(d∘≤dP)(xf(F))2r∣F∣∙≤dPd∘≤dP(xf(F)dP(xf(F))2r∣F∣∙)≤dPd∘≤dP(2r∣F∣)<r∣F∣−1.
Thus we have yF≤dP(xf(F))2r∣F∣∙ satisfying xf(F)dyF<r∣F∣−1. We claim that the net (yF) obtained in this way is <d-increasing. Indeed, if F⫋G then we can take positive ϵ<r∣G∣−1−xf(G)dyG. If y∈X satisfies yGdy<ϵ then
[TABLE]
So (yG<ϵd)⊆(xf(F))2r∣F∣∙⊆(yF≤d), i.e. yF<dyG, proving the claim. Thus Y={yF:F∈F(Λ)} is <d-directed. Also F⫋G implies
[TABLE]
so (xλ)≤dY. And for λ≻f(F), xλ∈(xf(F))r∣F∣∙⊆(xf(F))2r∣F∣∙ so yF≤dxλ and hence Y≤d(xλ).
∎
Next we consider a different condition on balls leading to an interpolation condition involving the function Fd defined on F(X)×X by
[TABLE]
So Fd is just the restriction to finite subsets of Pd from (10.8).
Proposition \theprp.
Every open lower d-ball is ≤d-directed if and only if
[TABLE]
Proof.
Assume every ball x∙r is ≤d-directed. Then, for all finite F⊆X, x∈X and r>Fdx, i.e. F⊆x∙r, we have y∈x∙r with F≤dy so ≤Fd∘d≤Fd. Conversely, if ≤Fd∘d≤Fd and we have finite F⊆x∙r then Fdx<r so we have y with F≤dy and ydx<r, i.e. y∈x∙r.
∎
Again C0(X)+ with fdg=supx∈X(f(x)−g(x))+ satisfies this condition, while any metric space with at least two points does not. We weaken the condition slightly and add a completeness assumption in the following, based on [Bic18, Theorem 2]. Note that the result is immediate when d is a transitive relation <, as <-max-completeness then implies that we can set each yn to be the <-maximum of any <-increasing subnet of (xλ).
Theorem 11.2**.**
If d is a distance, ≤Fd∘d≤Fd and, moreover, X is ≤d-(d-max)-complete then
[TABLE]
Proof.
The basic idea of the proof will be to replace a given d-pre-Cauchy net by one indexed by F(Λ) and then further replace this by a ≤d-increasing net. The resulting limit will still be off the mark by a small amount, so we actually have to consider countably many tails of F(Λ) and replace each of the corresponding subnets by ≤d-increasing nets.
First note ≤Fd∘d≤Fd is equivalent to saying that
[TABLE]
is below the identity function on [0,∞]. This, in turn, is equivalent to saying that the f-image of [0,r) is contained in [0,r), for all r∈(0,∞). In fact, it suffices that there are arbitrarily small such r.
So we assume we have rn↓0 with f[0,rn)⊆[0,rn), for all n∈N. Then, for each n, we have positive rnm↑rn (i.e. limmrnm=rn) with f(rnm)<rnm+1, for all m∈N. Taking f(rn0)=0 below, set
[TABLE]
Again take d-pre-Cauchy (xλ)⊆X. Again, if necessary, we can replace (xλ) with a d-Cauchy net, by Theorem 7.1 (1), and the conclusion of the theorem will be preserved, by Theorem 7.1 (3) (noting that, as d is a distance, any d-pre-Cauchy net (xλ) is d-pre-Cauchy, by (2.3), so dxλ converges hence any subnet also converges to the same limit). Define f:F(Λ)→Λ recursively so f({λ})=λ, for all λ∈Λ, f(E)≺f(F), for all F∈F(Λ) with ∣F∣>1 and all E⫋F, and
[TABLE]
For any n∈N, let Λn={F∈F(Λ):∣F∣>n} and define (yFn)F∈Λn recursively as follows. For ∣F∣=n+1, let yFn=xf(F) so if F⫋G then
[TABLE]
For ∣G∣=n+2, let Y={yFn:F⫋G and ∣F∣=n+1}. Now
[TABLE]
so we can take yGn with
[TABLE]
As xf(G)dxf(H)<ϵn2, whenever G⫋H and ∣G∣=n+2,
[TABLE]
Note that we also have
[TABLE]
Thus if ∣H∣=n+3 and Z=⋃{{yGn,xf(G)}:G⫋H and ∣G∣=n+2},
[TABLE]
Thus we can take yHn with Z≤dyHn and
[TABLE]
Continuing in this way we obtain ≤d-increasing (yFn) with yFndxf(G)<rn and xf(F)≤dyGn, for all F∈Λn and F⫋G.
As X is ≤d-d-max-complete, we can take yn=d-maxFyFn. Thus xf(F)≤dyn, for all F∈Λn. As (xλ) is d-Cauchy, Theorem 7.1 (2) implies that xλd converges. As (xf(F))F∈Λn is a subnet of (xλ), for each n∈N we have
[TABLE]
Thus
[TABLE]
d(yn)≤d(xλ). Also, for any m,n∈N and all sufficiently large H, G and F, specifically H⫌G⫌F∈Λm∨n,
[TABLE]
By (10.4), ym=d-supFyFm so ymdyn≤limFyFmdyn≤rm and hence ymd∨yn≤rm∧n. As rn→0, this shows that (yn) is d∨-Cauchy. Also, as (xλ) is d-Cauchy, dxλ converges, by Theorem 7.1 (3), so
[TABLE]
Another natural interpolation condition involves the symmetrization:
[TABLE]
Yet again this is satisfied by C0(X)+ where fdg=supx∈X(f(x)−g(x))+. More interestingly, even in non-commutative C*-algebras we have the weaker uniform interpolation condition d∨∘≤d⪷d on the positive unit ball, where adb=∥(a−b)+∥ (see [BV18, Theorem 2.6]). For the result itself, based on [Bic18, Theorem 3], it again suffices to consider an even weaker condition, this time involving a modification of e ∘≤d defined by
[TABLE]
In particular, note (e∘Φd)≤(e∘≤d). Also note that when d is a metric, Y below must be a singleton set {x} with xλ→x. In this case, the result is really just saying that limits coincide for uniformly equivalent metrics.
Theorem 11.3**.**
If d and e are distances, X is e∘-complete, e∘Φd⪷d and d,dop⪷e then
[TABLE]
Proof.
Given d-pre-Cauchy (xλ), we may again take a subnet indexed by F(Λ) if necessary and assume we have nets (sλ),(tλ)⊆(0,∞) such that
[TABLE]
For each λ, we define γλn and xλn recursively so that
[TABLE]
First set γλ1=λ and xλ1=xλ. For n∈N, take γλn+1≻γλn with de∘Φd(sγλn+1),sγλn+1<2−ntλ. As xλndxγλn+1≤xλndxγλn+xγλndxγλn+1<sγλn,
[TABLE]
so we can take xλn+1 with xλnexλn+1<21−ntλ and
[TABLE]
Note the right side above is positive by (11.3) (with γλn+1 in place of λ). Thus the recursion may continue.
For each λ, xλnexλn+1<21−ntλ so (xλn)n∈N is e-Cauchy. As X is e∘-complete, we have yλ∈X with limnxλneyλ=0, by (8.15), and hence limnyλdxλn=0, as dop⪷e. Now
[TABLE]
So Y≤d(xλ) for Y={yλ:λ∈Λ}. As xλ=xλ1 and xλnexλn+1<21−ntλ, xλeyλ≤2tλ→0. Thus xλdyλ→0, as d⪷e. Now
[TABLE]
Thus (xλ)≤dY and hence Y is d-directed, by (9.10).
∎
Replacing Φd with Φd, we get ≤d-directed subsets from d-pre-Cauchy sequences (rather than d-directed subsets from d-pre-Cauchy nets). In fact, as the subset Y is countable, it could even be replaced with a cofinal increasing sequence. Indeed, this is how Y is constructed in the proof, which is based on the argument given in [Bic18, Theorem 4.5].
Theorem 11.4**.**
If d and e are distances, X is e∘-complete, e∘Φd⪷d and d,dop⪷e then e∘Φd=e∘≤d and
[TABLE]
Proof.
First we prove e∘Φd=e∘≤d. For any x,y∈X and ϵ>0, take ϵn↓0 with de∘Φd(ϵn)<2−nϵ, for all n∈N. Now take z1∈X with xez1<x(e∘Φd)y+ϵ and z1dy<ϵ1. Thus
[TABLE]
and we can take z2∈X such that z1ez2<21ϵ and z2dy<ϵ2. Continuing in this way we obtain a sequence (zn)⊆X such that, for all n∈N,
[TABLE]
As e is a distance and X is e∘-complete, (8.15) yields znez→0, for some z∈X, so
[TABLE]
Also zdy≤zdzn+zndy≤zdzn+ϵn→0, as dop⪷e and znez→0, so z≤dy. As ϵ>0 was arbitrary, (e∘≤d)≤(e∘Φd). The reverse inequality is immediate.
Now take (snm),(tnm)⊆(0,∞) such that, for all m,n∈N,
[TABLE]
(define (s1m)m∈N first then (t1m)m∈N, (s2m)m∈N etc., also note that the top of ed above is d, not d). Take a subsequence of the given d-pre-Cauchy (xn) with xndxn+1<tn1, for all n, and define ynm with ynmdyn+1m<tnm, for all m and n, recursively as follows. First let yn1=xn, for all n. Assume ynm is defined for all n and fixed m. For each n, we can take ynm+1≤dyn+1m with ynmeynm+1<snm+1 as
[TABLE]
Then the recursion may continue because
[TABLE]
For all m,n∈N, ynmeynm+1<snm+1<2−m−1−n<2−m−n so, as X is e∘-complete, we have yn∈X with limmynmeyn=0. As d,dop⪷e and ynm+1≤dyn+1m,
[TABLE]
i.e. yn≤dyn+1 so Y={yn:n∈N} is ≤d-directed. Also, again using the fact that e is a distance, we have
[TABLE]
This together with d⪷e and the fact that (xn) is d-pre-Cauchy yields
[TABLE]
so (xn)≤dY. Likewise, using dop⪷e instead and the fact d is a distance,
[TABLE]
so Y≤d(xn).
∎
As promised, we can now show that d∘∙-completeness follows from d-max-completeness (or even slightly weaker notions) under various additional interpolation, completeness and separability conditions.
Corollary \thecor.
X* is d∘∙-complete if d and e are distances satisfying any of the following (d-R-T-complete means d-R-complete and T-complete).*
[TABLE]
Proof.
Take d-Cauchy (xλ).
By Theorem 11.1, we have <d-directed Y such that Y≡d(xλ) and hence dY=d(xλ), by (2.3) and § 9. By <d-(d-max)-completeness and (10.5), we have x∈X with dx=dY=d(xλ) so xλ→∘∙x, by (8.12). Thus X is d∘∙-complete.
By Theorem 11.2, we have d∨-Cauchy (yn) with d(xλ)=d(yn). By d∘∨-completeness and (8.15), we have x∈X with ynd∨x→0 and hence yn→∘∙x, by (8.10). Thus dx=d(yn)=d(xλ) and hence xλ→∘∙x, by (8.12). Thus X is d∘∙-complete.
By Theorem 11.3, we have d-directed Y≡d(xλ) and hence dY=d(xλ), by (2.3) and § 9. By d-max-completeness and (10.5), we have x∈X with dx=dY=d(xλ) so xλ→∘∙x, by (8.5). Thus X is d∘∙-complete.
By Theorem 11.3, we have d-directed Y≡d(xλ). By (9.11), we have (xn′)n∈N≡dY. By Theorem 11.4, we have ≤d-directed Y′≡d(xn′) and hence dY′=d(xλ), by (2.3) and § 9. By ≤d-(d-max)-completeness, we have x∈X with dx=dY′=d(xλ), i.e. xλ→∘∙x. Thus X is d∘∙-complete.∎
