Endpoints in 𝑇₀-Quasimetric Spaces: Part II

Abstract

We continue our work on endpoints and startpoints in 𝑇₀-quasimetric spaces. In particular we specialize some of our earlier results to the case of two-valued 𝑇₀-quasimetrics, that is, essentially, to partial orders. For instance, we observe that in a complete lattice the startpoints (resp., endpoints) in our sense are exactly the completely join-irreducible (resp., completely meet-irreducible) elements. We also discuss for a partially ordered set the connection between its Dedekind-MacNeille completion and the 𝑞-hyperconvex hull of its natural 𝑇₀-quasimetric space.

1. Introduction

During his investigations on the hyperconvex (or injective) hull of a metric space Isbell [1] introduced the concept of an endpoint of a metric space and proved among other things that the hyperconvex hull of a compact metric space is equal to the hyperconvex hull of the subspace consisting of its endpoints (cf. also [2, 3]). A theory for 𝑇₀-quasimetric spaces similar to the one for metric spaces due to Isbell can be developed (see, e.g., [4, 5]). In particular each 𝑇₀-quasimetric space has a 𝑞-hyperconvex (or injective) hull. For instance, it turns out that the hyperconvex hull of a metric space 𝑋 is isometric to the largest metric subspace containing the canonical copy of 𝑋 in the 𝑞-hyperconvex hull of 𝑋 (see [5, Theorem 6]).

In [6] the authors defined the concept of an endpoint in an arbitrary 𝑇₀-quasimetric space. In the quasimetric context it turned out to be natural to consider also the dual concept of an endpoint, which we called a startpoint.

Improving on a result from [6] in this note we will show that for any join compact 𝑇₀-quasimetric space (𝑋,𝑑) the set of the endpoints (resp., startpoints) of 𝑋 is equal to the set of the endpoints (resp., startpoints) of its 𝑞-hyperconvex hull (𝑄_𝑋,𝐷).

We also specialize some of our earlier results in [6] to two-valued 𝑇₀-quasimetric spaces. It is well known that they are, essentially, the partially ordered sets and that in the category of partially ordered sets the injective hull coincides with the Dedekind-MacNeille completion (see, e.g., [5, 7, 8]). We will observe that in the case of a complete lattice our startpoints (resp., endpoints) turn out to be exactly the completely join-irreducible (resp., the completely meet-irreducible) elements. We also discuss for a partially ordered set in some detail the connection between its Dedekind-MacNeille completion and the 𝑞-hyperconvex hull of its natural 𝑇₀-quasimetric space. Our results can, for instance, be used to analyze the similarity between the following result due to Isbell [1]: A compact injective metric space 𝑌 has a smallest closed subset 𝐵 such that the hyperconvex hull of 𝐵 is equal to 𝑌; and the following well-known result from order theory (see, e.g., [9, Theorem 7.42]): A lattice 𝐿 with no infinite chains is order isomorphic to the Dedekind-MacNeille completion of the partially ordered set 𝒥(𝐿)∪ℳ(𝐿), where 𝒥(𝐿) denotes the set of join-irreducible elements of 𝐿 and ℳ(𝐿) denotes the set of meet-irreducible elements of 𝐿. Furthermore 𝒥(𝐿)∪ℳ(𝐿) is the smallest subset of 𝐿 which is both join- and meet-dense in 𝐿.

2. Preliminaries

In this section we first recall some of the basic definitions from asymmetric topology needed to read this paper. Then we recall some fundamental facts of the theory of the 𝑞-hyperconvex hull of a 𝑇₀-quasimetric space.

Definition 1. Let 𝑋 be a set and 𝑑∶𝑋×𝑋→[0,∞) a function. (Here [0,∞) denotes the set of the nonnegative reals.) Then 𝑑 is quasipseudometric on 𝑋 if(a)𝑑(𝑥,𝑥)=0 whenever 𝑥∈𝑋, and(b)𝑑(𝑥,𝑧)≤𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧) whenever 𝑥,𝑦,and𝑧∈𝑋.
We will say that (𝑋,𝑑) is a 𝑇₀-quasimetric space provided that 𝑑 also satisfies the following condition: for each 𝑥,𝑦∈𝑋, 𝑑(𝑥,𝑦)=0=𝑑(𝑦,𝑥) implies that 𝑥=𝑦.

Let 𝑑 be quasipseudometric on a set 𝑋. Then 𝑑⁻¹∶𝑋×𝑋→[0,∞) defined by 𝑑⁻¹(𝑥,𝑦)=𝑑(𝑦,𝑥) whenever 𝑥,𝑦∈𝑋 is also quasipseudometric, called the conjugate quasipseudometric of 𝑑. Observe that if 𝑑 is a 𝑇₀-quasimetric on 𝑋, then 𝑑^𝑠=max{𝑑,𝑑⁻¹}=𝑑∨𝑑⁻¹ is a metric on 𝑋.

Given 𝑥∈𝑋 and a nonnegative real number 𝑟 we set 𝐶_𝑑(𝑥,𝑟)={𝑦∈𝑋∶𝑑(𝑥,𝑦)≤𝑟}. Note that this set is 𝜏(𝑑⁻¹)-closed, where 𝜏(𝑑) is the topology having the balls 𝐵_𝜖(𝑥)={𝑦∈𝑋∶𝑑(𝑥,𝑦)<𝜖} with 𝑥∈𝑋 and 𝜖>0 as basic (open) sets.

A map 𝑓∶(𝑋,𝑑)→(𝑌,𝑒) between quasipseudometric spaces (𝑋,𝑑) and (𝑌,𝑒) is called isometric provided that 𝑑(𝑥,𝑦)=𝑒(𝑓(𝑥),𝑓(𝑦)) whenever 𝑥,𝑦∈𝑋. Each isometric map with a 𝑇₀-quasimetric domain is a one-to-one map.

Furthermore a map 𝑓∶(𝑋,𝑑)→(𝑌,𝑒) between quasipseudometric spaces (𝑋,𝑑) and (𝑌,𝑒) is called nonexpansive provided that 𝑒(𝑓(𝑥),𝑓(𝑦))≤𝑑(𝑥,𝑦) whenever 𝑥,𝑦∈𝑋.

Given two real numbers 𝑎 and 𝑏 we will write 𝑎−̇𝑏 for max{𝑎−𝑏,0}, which we will also denote by (𝑎−𝑏)∨0. Note that 𝑢(𝑥,𝑦)=𝑥−̇𝑦 with 𝑥,𝑦∈ℝ defines the standard 𝑇₀-quasimetric on the set ℝ of the reals.

Given a 𝑇₀-quasimetric space (𝑋,𝑑), we recall that the specialization (partial) order ≤_𝑑 of 𝑑 is defined as follows: for each 𝑥,𝑦∈𝑋, set 𝑥≤_𝑑𝑦 if and only if 𝑑(𝑥,𝑦)=0.

For further basic concepts used from the theory of asymmetric topology we refer the reader to [10–12].

Many facts about hyperconvexity in metric spaces can be found in [13–15]. Connections between that theory and order theory are explored in [7, 16]. Throughout we will assume familiarity of the reader with the results of [6].

We next recall some facts mainly from [4] belonging to the theory of the 𝑞-hyperconvex hull of a 𝑇₀-quasimetric space (see also [5, 8, 17–19] for some related investigations).

Let (𝑋,𝑑) be a 𝑇₀-quasimetric space. We will say that a function pair 𝑓=(𝑓₁,𝑓₂) on (𝑋,𝑑), where 𝑓_𝑖∶𝑋→[0,∞)(𝑖=1,2) is ample provided that 𝑑(𝑥,𝑦)≤𝑓₂(𝑥)+𝑓₁(𝑦) whenever 𝑥,𝑦∈𝑋.

Let 𝑃_𝑋 denote the set of all ample function pairs on (𝑋,𝑑). (In such situations we may also write 𝑃_(𝑋,𝑑) in cases where 𝑑 is not obvious.) For each 𝑓,𝑔∈𝑃_𝑋 we set 𝐷𝑓,𝑔=sup_𝑥∈𝑋𝑓₁(𝑥)−̇𝑔₁(𝑥)∨sup_𝑥∈𝑋𝑔₂(𝑥)−̇𝑓₂(𝑥).(1)

Then 𝐷 is an extended (if we replace in the definition of a quasipseudometric [0,∞) by [0,∞] we obtain the definition of an extended quasipseudometric. Of course, the triangle inequality for extended quasipseudometrics is interpreted in the self-explanatory way. (Indeed some authors prefer to work with extended quasipseudometrics throughout; see, e.g., [8]). For the purpose of this paper however real-valued 𝑇₀-quasimetrics seem to be more appropriate; cf. Section 7) 𝑇₀-quasimetric on 𝑃_𝑋.

We will call a function pair 𝑓 minimal on (𝑋,𝑑) (among the ample function pairs on (𝑋,𝑑)) if it is ample and whenever 𝑔 is ample on (𝑋,𝑑), and for each 𝑥∈𝑋 we have 𝑔₁(𝑥)≤𝑓₁(𝑥) and 𝑔₂(𝑥)≤𝑓₂(𝑥) (for any function pairs 𝑓 and 𝑔 satisfying this relation we will write 𝑔≤𝑓); then 𝑔=𝑓. It is well known that Zorn’s Lemma implies that below each ample function pair there is a minimal ample pair (cf., e.g., [2, 20]). By 𝑄_𝑋 we will denote the set of all minimal ample pairs on (𝑋,𝑑) equipped with the restriction of 𝐷 to 𝑄_𝑋×𝑄_𝑋, which we will also denote by 𝐷. Recall that 𝐷 is a (real-valued) 𝑇₀-quasimetric on 𝑄_𝑋×𝑄_𝑋 [4, Remark 6].

Furthermore 𝑓=(𝑓₁,𝑓₂)∈𝑄_𝑋 if and only if the following equations (∗) are satisfied: 𝑓₁(𝑥)=sup𝑑𝑦,𝑥−̇𝑓₂𝑦,𝑓∶𝑦∈𝑋₂(𝑥)=sup𝑑𝑥,𝑦−̇𝑓₁𝑦(∗)∶𝑦∈𝑋 whenever 𝑥∈𝑋 (cf. [21, Remark 2]). In particular note that such pairs are ample on (𝑋,𝑑).

Obviously the second component 𝑓₂ of a minimal ample pair (𝑓₁,𝑓₂) on (𝑋,𝑑) satisfies the following equation (∗∗): 𝑓₂(𝑥)=sup_𝑦∈𝑋𝑑𝑥,𝑦−̇sup_{𝑦′∈𝑋}𝑑𝑦^′,𝑦−̇𝑓₂𝑦^′(∗∗) whenever 𝑥∈𝑋.

Given any real-valued function 𝑓∶𝑋→[0,∞), satisfying (∗∗), we can set 𝑓₁(𝑥)∶=sup{𝑑(𝑦,𝑥)−̇𝑓(𝑦)∶𝑦∈𝑋} whenever 𝑥∈𝑋.

One readily checks that (𝑓₁,𝑓) is an ample function pair on (𝑋,𝑑).

Furthermore, of course, 𝑓₁(𝑥)𝑑𝑦,𝑓(𝑥)𝑑=sup𝑦,𝑥−̇𝑓∶𝑦∈𝑋=sup𝑥,𝑦−̇𝑓₁𝑦∶𝑦∈𝑋(2) whenever 𝑥∈𝑋. Hence (𝑓₁,𝑓) is a minimal ample pair on (𝑋,𝑑), and thus (𝑓₁,𝑓)∈𝑄_𝑋.

Hence (∗∗) characterizes exactly those functions 𝑓∶𝑋→[0,∞) that are second component of minimal ample pairs on (𝑋,𝑑) (cf., e.g., [5, 8] concerning the underlying Isbell conjugation adjunction; see also [22]). Of course, an analogous result holds for the first component of minimal ample pairs on (𝑋,𝑑).

It is known (see [4, Lemma 3]) that 𝑓∈𝑄_𝑋 implies that 𝑓₁(𝑥)−𝑓₁(𝑦)≤𝑑(𝑦,𝑥) and 𝑓₂(𝑥)−𝑓₂(𝑦)≤𝑑(𝑥,𝑦) whenever 𝑥,𝑦∈𝑋.

Moreover sup_𝑥∈𝑋(𝑓₁(𝑥)−̇𝑔₁(𝑥))=sup_𝑥∈𝑋(𝑔₂(𝑥)−̇𝑓₂(𝑥)) whenever 𝑓,𝑔∈𝑄_𝑋 (cf., [4, Lemma 7]).

For each 𝑥∈𝑋 we can define the minimal function pair 𝑓_𝑥𝑦=𝑑𝑥,𝑦,𝑑𝑦,𝑥(3) whenever 𝑦∈𝑋 on (𝑋,𝑑). The map 𝑒 defined by 𝑥↦𝑓_𝑥 whenever 𝑥∈𝑋 defines an isometric embedding of (𝑋,𝑑) into (𝑄_𝑋,𝐷) (see [4, Lemma 1]).

Then (𝑄_𝑋,𝐷) is called the 𝑞-hyperconvex hull of (𝑋,𝑑). A 𝑇₀-quasimetric space 𝑋 is said to be 𝑞-hyperconvex if 𝑓∈𝑄_𝑋 implies that there is an 𝑥∈𝑋 such that 𝑓=𝑓_𝑥 (cf. [4, Corollary 4]). An intrinsic characterization of 𝑞-hyperconvexity of a 𝑇₀-quasimetric space (𝑋,𝑑) can, for instance, be found in [4, Definition 2]: a 𝑇₀-quasimetric space (𝑋,𝑑) is 𝑞-hyperconvex if and only if, given 𝐴⊆𝑋 and families of nonnegative reals (𝑟_𝑥)_𝑥∈𝐴 and (𝑠_𝑥)_𝑥∈𝐴 such that 𝑑(𝑥,𝑦)≤𝑟_𝑥+𝑠_𝑦 whenever 𝑥,𝑦∈𝐴, we have that ⋂_𝑥∈𝐴(𝐶_𝑑(𝑥,𝑟_𝑥)∩𝐶_𝑑⁻¹(𝑥,𝑠_𝑥))≠∅ (see [4, Remark 2]).

Note also that 𝐷(𝑓,𝑓_𝑥)=𝑓₁(𝑥) and 𝐷(𝑓_𝑥,𝑓)=𝑓₂(𝑥) whenever 𝑥∈𝑋 and 𝑓∈𝑄_𝑋 [4, Lemma 8].

The following important result (see [4, Remark 7]) is best understood as a kind of density of 𝑋 in 𝑄_𝑋. For any 𝑦₁,𝑦₂∈𝑄_𝑋, we have that 𝐷𝑦₁,𝑦₂=sup𝐷𝑓_𝑥1,𝑓_𝑥2−𝐷𝑓_𝑥1,𝑦₁−𝐷𝑦₂,𝑓_𝑥2∨0∶𝑥₁,𝑥₂∈𝑋.(4)

Our first example shows that in that formula the step of taking the supremum with 0 cannot be avoided in general.

Example 2. Let 𝑎,𝑏∈[0,∞) be such that 𝑎,𝑏>0, and let 𝑌=[0,𝑎]×[0,𝑏]. Set 𝐷𝛼₁,𝛼₂,𝛽₁,𝛽₂=𝛼₁−̇𝛽₁∨𝛼₂−̇𝛽₂(5) whenever (𝛼₁,𝛼₂),(𝛽₁,𝛽₂)∈𝑌. It is known that 𝑌 can be identified with the 𝑞-hyperconvex hull of the subspace 𝑋={(𝑎,0),(0,𝑏)} of 𝑌 (see [6, Example 4]).
Furthermore [𝐷),(),(),(∨[𝐷),(),(),(=[]∨[]),(.((𝑎,00,𝑏))−𝐷((𝑎,00,0))−𝐷((𝑎,𝑏0,𝑏))]((0,𝑏𝑎,0))−𝐷((𝑏,00,0))−𝐷((𝑎,𝑏𝑎,0))]𝑎−𝑎−𝑎𝑏−𝑏−𝑏≠0=𝐷((0,0𝑎,𝑏))(6)

3. The Concept of Collinearity in Quasipseudometric Spaces

The following definition was given in [6] (cf. [3]). Let (𝑋,𝑑) be a quasipseudometric space.(1)A finite sequence (𝑥₁,𝑥₂,…,𝑥_𝑛) in 𝑋 is called collinear in (𝑋,𝑑) provided that 𝑖<𝑗<𝑘≤𝑛 implies that 𝑑(𝑥_𝑖,𝑥_𝑘)=𝑑(𝑥_𝑖,𝑥_𝑗)+𝑑(𝑥_𝑗,𝑥_𝑘).(2)An element 𝑥∈𝑋 is called an endpoint of (𝑋,𝑑) provided that there exists an element 𝑦 in (𝑋,𝑑) such that 𝑑(𝑦,𝑥)>0 and such that for any 𝑧∈𝑋 collinearity of (𝑦,𝑥,𝑧) in (𝑋,𝑑) implies that 𝑥=𝑧. We will say that 𝑦witnesses that 𝑥 is an endpoint.(3)An element 𝑥∈𝑋 is called a startpoint of (𝑋,𝑑) if it is an endpoint of (𝑋,𝑑⁻¹).

Let us finally recall that a quasipseudometric space (𝑋,𝑑) is called join compact provided that 𝜏(𝑑^𝑠) is compact.

The next result says intuitively that the points in the remainder of the 𝑞-hyperconvex hull 𝑄_𝑋 of a join compact 𝑇₀-quasimetric space 𝑋 lie between the points of 𝑋.

Proposition 3. Let (𝑋,𝑑) be a join compact 𝑇₀-quasimetric space, and let 𝑒∶(𝑋,𝑑)→(𝑄_𝑋,𝐷) be the canonical isometric embedding of (𝑋,𝑑) into its 𝑞-hyperconvex hull (𝑄_𝑋,𝐷). Consider any 𝑓∈𝑄_𝑋⧵𝑒(𝑋). Then there are a startpoint 𝑠 and an endpoint 𝑒 in (𝑋,𝑑) such that (𝑠,𝑓,𝑒) is collinear in (𝑄_𝑋,𝐷).

Proof. As stated above, for 𝑦∈𝑋 we will identify 𝑓_𝑦 with 𝑦.
Fix 𝑥∈𝑋. Since (𝑄_𝑋,𝐷) is a 𝑇₀-space, we have that 𝐷(𝑓,𝑥)>0 or 𝐷(𝑥,𝑓)>0. We consider only the first case. The second one is analogous. By [6, Lemma 2] there is 𝑦∈𝑋 such that (𝑦,𝑓,𝑥) is collinear in (𝑄_𝑋,𝐷).
Then 𝐷(𝑦,𝑥)≥𝐷(𝑓,𝑥)>0. Therefore by [6, Corollary 3] there are a startpoint 𝑠 and an endpoint 𝑒 in (𝑋,𝑑) such that (𝑠,𝑦,𝑥,𝑒) is collinear in (𝑄_𝑋,𝐷).
It follows that 𝐷(𝑠,𝑓)+D(𝑓,𝑒)≤(𝐷(𝑠,𝑦)+𝐷(𝑦,𝑓))+(𝐷(𝑓,𝑥)+𝐷(𝑥,𝑒))=𝐷(𝑠,𝑦)+𝐷(𝑦,𝑥)+ 𝐷(𝑥,𝑒)=(𝐷(𝑠,𝑦)+𝐷(𝑦,𝑥))+𝐷(𝑥,𝑒) = 𝐷(𝑠,𝑥)+𝐷(𝑥,𝑒)=𝐷(𝑠,𝑒), and hence (𝑠,𝑓,𝑒) is collinear in (𝑄_𝑋,𝐷).

Lemma 4. Let (𝑋,𝑑) be a 𝑇₀-quasimetric space. If 𝑥 is an endpoint with witness 𝑦 and (𝑧,𝑦,𝑥) is collinear in (𝑋,𝑑), then 𝑧 is also a witness that 𝑥 is an endpoint in (𝑋,𝑑). Similarly, if 𝑥 is a startpoint with witness 𝑦 and (𝑥,𝑦,𝑧) is collinear in (𝑋,𝑑), then 𝑧 is also a witness that 𝑥 is a startpoint in (𝑋,𝑑).

Proof. Assume that 𝑥 is an endpoint with witness 𝑦 and that (𝑧,𝑦,𝑥) is collinear in (𝑋,𝑑). Note that 𝑑(𝑧,𝑥)≥𝑑(𝑦,𝑥)>0. Suppose that for some 𝑎∈𝑋,(𝑧,𝑥,𝑎) is collinear in (𝑋,𝑑). Then inequalities 𝑑(𝑧,𝑎)≤𝑑(𝑧,𝑦)+𝑑(𝑦,𝑎)≤𝑑(𝑧,𝑦)+𝑑(𝑦,𝑥)+𝑑(𝑥,𝑎)=𝑑(𝑧,𝑥)+𝑑(𝑥,𝑎)=𝑑(𝑧,𝑎) imply by subtracting 𝑑(𝑧,𝑦) that 𝑑(𝑦,𝑎)=𝑑(𝑦,𝑥)+𝑑(𝑥,𝑎) (cf., [6, Lemma 1]); hence (𝑦,𝑥,𝑎) is collinear in (𝑋,𝑑), and thus 𝑥=𝑎 by our assumption. Therefore 𝑧 witnesses that 𝑥 is an endpoint of (𝑋,𝑑). The dual result is proved analogously.

Example 5. Let (𝑋,𝑑) be a join compact 𝑇₀-quasimetric space with 𝑦₁,𝑦₂∈𝑋 such that 𝑑(𝑦₁,𝑦₂)>0.
According to the proofs of [6, Proposition 3, Corollary 3] there exist a startpoint 𝑠 in (𝑋,𝑑) with witness 𝑦₂ and an endpoint 𝑒 in (𝑋,𝑑) with witness 𝑠 such that (𝑠,𝑦₁,𝑦₂,𝑒) is collinear in (𝑋,𝑑). By Lemma 4 it follows that both 𝑠 witnesses that 𝑒 is an endpoint and 𝑒 witnesses that 𝑠 is a startpoint in (𝑋,𝑑).

We next prove the result mentioned in Section 1.

Proposition 6. Let (𝑋,𝑑) be a join compact 𝑇₀-quasimetric space. Then (𝑄_𝑋,𝐷) has exactly the same endpoints and startpoints as (𝑋,𝑑).

Proof. In [6, Proposition 4] it was shown that each endpoint of (𝑋,𝑑) is an endpoint of (𝑄_𝑋,𝐷). In fact, it was proved that if 𝑥 is an endpoint in (𝑋,𝑑) with witness 𝑦∈𝑋, then 𝑓_𝑦 witnesses that 𝑓_𝑥 is an endpoint in (𝑄_𝑋,𝐷).
Here we will show that each endpoint of (𝑄_𝑋,𝐷) is an endpoint of (𝑋,𝑑).
Suppose that 𝑓∈𝑄_𝑋 witnesses that 𝑔∈𝑄_𝑋 is an endpoint in (𝑄_𝑋,𝐷). Then 𝐷(𝑓,𝑔)>0. Thus by [20, Proposition 5] there are sequences 𝑥_𝑛,𝑦_𝑛∈𝑋 such that the increasing sequence (𝐷(𝑥_𝑛,𝑦_𝑛)−𝐷(𝑥_𝑛,𝑓)−𝐷(𝑔,𝑦_𝑛))_𝑛∈ℕ converges to 𝐷(𝑓,𝑔) with respect to the usual topology on ℝ. (Note that here and in the following, for any 𝑥∈𝑋, we will identify 𝑥 with 𝑓_𝑥.)
By join compactness of (𝑋,𝑑) indeed we have that there is a subsequence (𝑛_𝑘)_𝑘∈ℕ of (𝑛)_𝑛∈ℕ and 𝑥,𝑦∈𝑋 such that 𝑑^𝑠(𝑥_𝑛𝑘,𝑥)→0 and 𝑑^𝑠(𝑦_𝑛𝑘,𝑦)→0. Note that, for instance, |𝐷(𝑥_𝑛𝑘,𝑓)−𝐷(𝑥,𝑓)|≤𝑑^𝑠(𝑥_𝑛𝑘,𝑥) for any 𝑘∈ℕ.
Taking limits, therefore 𝐷(𝑓,𝑔)=𝐷(𝑥,𝑦)−𝐷(𝑥,𝑓)−𝐷(𝑔,𝑦), and thus 𝐷(𝑥,𝑦)=𝐷(𝑥,𝑓)+𝐷(𝑓,𝑔)+𝐷(𝑔,𝑦). Consequently (𝑓,𝑔,𝑦) is collinear in (𝑄_𝑋,𝐷), since 𝐷(𝑥,𝑓)+𝐷(𝑓,𝑔)+𝐷(𝑔,𝑦)=𝐷(𝑥,𝑦)≤𝐷(𝑥,𝑓)+𝐷(𝑓,𝑦)≤𝐷(𝑥,𝑓)+𝐷(𝑓,𝑔)+𝐷(𝑔,𝑦). Hence 𝐷(𝑥,𝑓)+𝐷(𝑓,𝑔)+𝐷(𝑔,𝑦)=𝐷(𝑥,𝑓)+𝐷(𝑓,𝑦), and the statement follows.
We conclude that 𝑔=𝑦, since 𝑓 witnesses that 𝑔 is an endpoint in (𝑄_𝑋,𝐷). Indeed then 𝐷(𝑥,𝑦)=𝐷(𝑥,𝑓)+𝐷(𝑓,𝑦), and (𝑥,𝑓,𝑔) is collinear in (𝑄_𝑋,𝐷). By Lemma 4 therefore 𝑥 is a witness belonging to 𝑋 that 𝑔=𝑦∈𝑋 is an endpoint in (𝑄_𝑋,𝐷). It follows that 𝑥 witnesses that 𝑦 is an endpoint in (𝑋,𝑑), because (𝑋,𝑑) is a subspace of (𝑄_𝑋,𝐷).
The assertion about startpoints is proved analogously.

4. 𝑇₀-Quasimetrics Induced by Partial Orders

Let (𝑋,≤) be a partially ordered set and 𝑦∈𝑋. In the following, we set ↑𝑦∶={𝑥∈𝑋∶𝑦≤𝑥} and ↓𝑦∶={𝑥∈𝑋∶𝑦≥𝑥}.

Given a partially ordered set (𝑋,≤), we equip it with the 𝑇₀-quasimetric 𝑑 given by setting, for all 𝑥,𝑦∈𝑋,𝑑(𝑥,𝑦)=0 if 𝑥≤𝑦 and 𝑑(𝑥,𝑦)=1 otherwise. Indeed 𝑑 is a 𝑇₀-ultraquasimetric, but we will not use this fact in this paper (see, e.g., [23]).

We will call 𝑑 the natural 𝑇₀-quasimetric of ≤. Note that a map 𝑓∶(𝑋,𝑑)→({0,1},𝑢) is nonexpansive if and only if 𝑓 is monotonically increasing. (Of course, here 𝑢 also denotes the restriction of 𝑢 to {0,1}².)

Observe that, if 𝑑 is the natural 𝑇₀-quasimetric induced by a partial order ≤, then 𝑑⁻¹ is the natural 𝑇₀-quasimetric induced by the partial order ≥.

As we noted in the Preliminaries section, with the help of the specialization order we can equip each 𝑇₀-quasimetric space (𝑋,𝑑) with a partial order. In [6, Proposition 1] we considered the following more sophisticated method.

Fix 𝑦∈𝑋. For 𝑎₁,𝑎₂∈𝑋, set 𝑎₁≤_𝑦𝑎₂ if (𝑦,𝑎₁,𝑎₂) is collinear in (𝑋,𝑑). Then ≤_𝑦 is a partial order on 𝑋 according to [6, Proposition 1]. Our next result shows that, if the 𝑇₀-quasimetric space (𝑋,𝑑) originates from a partial order ≤, then ≤_𝑦 can be readily described.

Example 7. Let (𝑋,≤) be a partially ordered set, and let 𝑑 be its natural 𝑇₀-quasimetric. Furthermore let 𝑦∈𝑋. Then ≤_𝑦 agrees with ≤ if the two relations are restricted to (↑𝑦)² or to (𝑋⧵↑𝑦)². Furthermore ≤_𝑦 restricted to (𝑋⧵↑𝑦)×↑𝑦 is empty, and ≤_𝑦 restricted to ↑𝑦×(𝑋⧵↑𝑦) agrees with the complement of ≤ restricted to ↑𝑦×(𝑋⧵↑𝑦). Hence ≤_𝑦 can be described as follows. For 𝑎,𝑏∈𝑋, we have 𝑎≤_𝑦𝑏 if Case 1 is 𝑎≤𝑏 and [(i) 𝑎,𝑏∈↑𝑦 or (ii) 𝑎,𝑏∈(𝑋⧵↑𝑦)] or if Case 2 is 𝑎≰𝑏,𝑎∈↑𝑦 and 𝑏∈(𝑋⧵↑𝑦). Note that obviously ≤_𝑦 is a linear order if ≤ is a linear order.

Let us consider two specific examples of this construction.

Example 8. (a) Let (𝑋,≤) be a partially ordered set with the smallest element 0. Then ≤₀=≤.
(b) Let (𝑋,≤) be a partially ordered set with the largest element 1. Then ≤₁ is obtained on 𝑋 by removing the top element 1 from 𝑋 and adding it as a new smallest element to 𝑋⧵{1}.
Let (𝑋,𝑑) be a 𝑇₀-quasimetric space. Given 𝑎,𝑐∈𝑋 we set that 𝐿_𝑎,𝑐∶={𝑏∈𝑋∶(𝑎,𝑏,𝑐) is collinear in (𝑋,𝑑)} (cf. [2, (4.2)]). We have that 𝑎,𝑐∈𝐿_𝑎,𝑐 and 𝐿_𝑎,𝑐 is obviously 𝜏(𝑑^𝑠)-closed.
In the special case of 𝑇₀-quasimetrics originating from partial orders the set 𝐿_𝑎,𝑐 can be described as follows.

Example 9. Let (𝑋,≤) be a partially ordered set with natural 𝑇₀-quasimetric 𝑑, and let 𝑎,𝑐∈𝑋. Then 𝐿_𝑎,𝑐 is equal to the interval [𝑎,𝑐]={𝑥∈𝑋∶𝑎≤𝑥≤𝑐} if 𝑎≤𝑐 and equal to ↑𝑎∪↓𝑐 if 𝑎≰𝑐.

Example 10. Let (𝑋,≤) be a partially ordered set and 𝑑 its natural 𝑇₀-quasimetric. Suppose that (𝑥₁,𝑥₂,𝑥₃,𝑥₄) is collinear in (𝑋,𝑑) and 𝑑(𝑥₂,𝑥₃)>0. Then the sequence (𝑑(𝑥_𝑖,𝑥_𝑖+1))_𝑖=1,2,3 is equal to 010.

Proof. Of course 𝑑(𝑥₂,𝑥₃)=1. Furthermore 𝑖∈{1,2} with 𝑑(𝑥_𝑖,𝑥_𝑖+1)=1=𝑑(𝑥_𝑖+1,𝑥_𝑖+2) is impossible, since (𝑥_𝑖,𝑥_𝑖+1,𝑥_𝑖+2) is collinear in (𝑋,𝑑). Thus we have that the sequence (𝑑(𝑥_𝑖,𝑥_𝑖+1))_𝑖=1,2,3 is alternating in {0,1}.

We are next going to illustrate the concepts of a startpoint (resp., endpoint) in the case of 𝑇₀-quasimetrics induced by partial orders. Let us first note that [6, Proposition 3] is useless for an infinite partially ordered set 𝑋 equipped with its natural 𝑇₀-quasimetric 𝑑, since 𝑑^𝑠 is the discrete metric, and therefore 𝜏(𝑑^𝑠) is compact if and only if 𝑋 is finite.

Lemma 11. Let (𝑋,≤) be a partially ordered set, 𝑑 its natural 𝑇₀-quasimetric, and 𝑥,𝑦∈𝑋. Then 𝑥 is a startpoint of (𝑋,𝑑) witnessed by 𝑦 if and only if 𝑥 is a minimal element in 𝑋⧵↓𝑦.

Proof. Suppose that 𝑥 is a startpoint of (𝑋,𝑑) witnessed by 𝑦∈𝑋. Then 𝑑(𝑥,𝑦)=1, and thus 𝑥≰𝑦. Furthermore for all 𝑎∈𝑋,𝑑(𝑎,𝑥)+𝑑(𝑥,𝑦)=𝑑(𝑎,𝑦) implies that 𝑥=𝑎. Hence 𝑎≤𝑥 and 𝑎∈𝑋⧵↓𝑦 imply that 𝑥=𝑎. Thus 𝑥 is minimal in 𝑋⧵↓𝑦.
Assume that 𝑥 is minimal in 𝑋⧵↓𝑦. Then 𝑑(𝑥,𝑦)>0. Suppose that, for some 𝑎∈𝑋,(𝑎,𝑥,𝑦) is collinear in (𝑋,𝑑). Then 𝑑(𝑎,𝑦)=𝑑(𝑎,𝑥)+𝑑(𝑥,𝑦)=𝑑(𝑎,𝑥)+1=1; thus 𝑎≰𝑦 and 𝑎≤𝑥. Therefore 𝑎=𝑥, since 𝑥 is minimal in 𝑋⧵↓𝑦. Hence 𝑥 is a startpoint in (𝑋,𝑑) witnessed by 𝑦.

Corollary 12. Let (𝑋,≤) be a partially ordered set, 𝑑 its natural 𝑇₀-quasimetric, and 𝑥,𝑦∈𝑋. Then 𝑥 is an endpoint in (𝑋,𝑑) witnessed by 𝑦 if and only if 𝑥 is a maximal element in 𝑋⧵↑𝑦.

We next illustrate Lemma 11 and its corollary by two simple examples.

Example 13. Let 𝑋 be a set having at least two points and equipped with the discrete order =. Then the 𝑇₀-quasimetric 𝑑 induced by = on 𝑋 is the discrete metric. Note that each point of 𝑋 is an endpoint and a startpoint in (𝑋,𝑑), witnessed by any other point.

Example 14. Let 𝑋 be a complete lattice, and let 𝑘∈𝑋 be compact (see [24, Definition I-1.1.] for the definition), but such that 𝑘≠0. Since 𝑘≠0, there is 𝑦∈𝑋 such that 𝑘≰𝑦. By Zorn’s Lemma, let 𝐶 be a maximal chain in 𝑋⧵↑𝑘 containing 𝑦. Set 𝑚=∨𝐶. Then 𝑚∈𝑋⧵↑𝑘, since 𝑘 is compact. Furthermore 𝑚∈𝐶, since 𝐶 is a maximal chain. Thus 𝑚 is maximal in 𝑋⧵↑𝑘. Consequently 𝑚 is an endpoint witnessed by 𝑘 in (𝑋,𝑑), where 𝑑 denotes the natural 𝑇₀-quasimetric of 𝑋.

The following definition can essentially be found in [24, Definition I-4.21]. Let 𝑋 be a partially ordered set. An element 𝑥∈𝑋 is called completely 𝑚-irreducible if either 𝑥 is maximal in 𝑋 but different from the largest element or the set ↑𝑥⧵{𝑥} has a least element, which will be denoted by 𝑥⁺. Dually one defines completely 𝑗-irreducible elements in 𝑋.

Proposition 15. (a) Each completely 𝑚-irreducible element 𝑥 in a partially ordered set (𝑋,≤) is an endpoint in (𝑋,𝑑), where 𝑑 denotes the natural 𝑇₀-quasimetric of 𝑋.
(b) Let 𝑋 be a complete lattice equipped with its natural 𝑇₀-quasimetric 𝑑 and 𝑥∈𝑋 (cf. [24, Remark I-4.23]). If 𝑥 is an endpoint of (𝑋,𝑑), then 𝑥 is completely 𝑚-irreducible.

Proof. (a) Suppose first that 𝑥 is maximal but not the largest element of 𝑋. Then there is 𝑦∈𝑋 such that 𝑦≰𝑥. Therefore 𝑥 is maximal in 𝑋⧵↑𝑦, and 𝑥 is an endpoint by Corollary 12.
Suppose now that ↑𝑥⧵{𝑥} has a least element 𝑥⁺. It will suffice by Corollary 1 to show that 𝑥 is maximal in 𝑋⧵↑𝑥⁺. So let 𝑎∈𝑋 with 𝑥<𝑎. Then 𝑎∈↑𝑥⧵{𝑥} and consequently 𝑥⁺≤𝑎. Hence 𝑎∉𝑋⧵↑𝑥⁺ and 𝑥 is maximal in 𝑋⧵↑𝑥⁺. Therefore 𝑥 is an endpoint in (𝑋,𝑑).
(b) For the convenience of the reader we include a proof, which follows [24, page 126]. Let 𝑘∈𝑋 witness that 𝑥 is an endpoint in (𝑋,𝑑). If 𝑥 is maximal, then 𝑥 is completely 𝑚-irreducible, as it cannot be the largest element in 𝑋, because 𝑘<𝑥̸.
If 𝑥 is not maximal in 𝑋, then ∅≠↑𝑥⧵{𝑥}⊆↑𝑘, since 𝑥 is maximal in 𝑋⧵↑𝑘 by Corollary 12. Hence 𝑥⁺∶=∧(↑𝑥⧵{𝑥}) exists and 𝑥⁺>𝑥, as 𝑥⁺≥𝑘. Thus ↑𝑥⧵{𝑥} has a least element. We have shown that 𝑥 is completely 𝑚-irreducible in either case.

Corollary 16. (a) Each completely 𝑗-irreducible element 𝑥 in a partially ordered set (𝑋,≤) is a startpoint in (𝑋,𝑑), where 𝑑 denotes the natural 𝑇₀-quasimetric of 𝑋.
(b) Let 𝑋 be a complete lattice equipped with its natural 𝑇₀-quasimetric 𝑑 and 𝑥∈𝑋 (cf., [24, Remark I-4.23]). If 𝑥 is a startpoint of (𝑋,𝑑), then 𝑥 is completely 𝑗-irreducible.

Recall that an element 𝑥 in a complete lattice 𝑋 is called completely join-irreducible if, for each subset 𝑆 of 𝑋,𝑥=∨𝑆 implies that 𝑥∈𝑆. Completely meet-irreducible elements are defined dually (see [9, Definition 10.26]).

Of course, in a complete lattice the completely 𝑗-irreducible elements are exactly the completely join-irreducible elements and the completely 𝑚-irreducible elements are exactly the completely meet-irreducible elements.

Corollary 17. Let 𝑋 be a complete lattice and 𝑑 its natural 𝑇₀-quasimetric. Then 𝑥∈𝑋 is a startpoint in (𝑋,𝑑) if and only if 𝑥 is completely join-irreducible. Similarly, 𝑥∈𝑋 is an endpoint in (𝑋,𝑑) if and only if 𝑥 is completely meet-irreducible in (𝑋,𝑑).

Example 18. In a partially ordered set 𝑋 that is not complete and is equipped with its natural 𝑇₀-quasimetric 𝑑, an endpoint of (𝑋,𝑑) need not be completely 𝑚-irreducible. As an example consider the partially ordered set 𝑃₄ from [9, page 169] (see Figure 1).
Our characterizations (Lemma 11 and Corollary 12) immediately yield the set of startpoints {𝑎,𝑏,𝑐,𝑒} and the set of endpoints {𝑏,𝑐,𝑑,𝑒}. In particular 𝑏 is not completely 𝑚-irreducible, although it is an endpoint.

Figure 1 

Hasse diagram of 𝑃4.

As usual, an element 𝑥 in a complete lattice 𝑋 will be called completely join-prime if 𝑥≤∨𝑆 for some subset 𝑆 of 𝑋 means that there is 𝑠∈𝑆 such that 𝑥≤𝑠. The concept of a completely meet-prime element is defined dually. It is easy to see that each completely join-prime element is completely join-irreducible and each completely meet-prime element is completely meet-irreducible (see [9, Definition 10.26]).

Example 19. Let 𝑋 be a complete lattice and 𝑑 its natural 𝑇₀-quasimetric. Furthermore let 𝑥∈𝑋 be completely join-prime. It follows that 𝑦=∨(𝑋⧵↑𝑥) is completely meet-prime. Furthermore one proves that ↓𝑦=𝑋⧵↑𝑥. Indeed (𝑥,𝑦) is a completely prime pair in the sense of [9, page 246]. We conclude that 𝑦 is an endpoint with witness 𝑥 in (𝑋,𝑑), since 𝑦 is obviously maximal in 𝑋⧵↑𝑥, and 𝑥 is a startpoint with witness 𝑦 in (𝑋,𝑑), since 𝑥 is minimal in 𝑋⧵↓𝑦.

We recall that a subset 𝐸 of a partially ordered set 𝑋 is called join-dense in 𝑋 provided that for each 𝑥∈𝑋 there exists 𝐸^′⊆𝐸 such that 𝑥=∨_𝑋𝐸^′ (see [9, page 53]). Dually one defines the concept of a meet-dense subset of a partially ordered set 𝑋.

It is known (see [24, Remark I-4.22]) that each meet-dense subset of 𝑋 contains all completely 𝑚-irreducible elements. Similarly each join-dense subset of 𝑋 contains all completely 𝑗-irreducible elements.

Proposition 20. Let 𝑋 be a partially ordered set and 𝑑 its natural 𝑇₀-quasimetric.(a)If 𝐸 is a join-dense subset of 𝑋, then all startpoints of (𝑋,𝑑) belong to 𝐸. Dually, if 𝐸 is a meet-dense subset in 𝑋, then all endpoints of (𝑋,𝑑) belong to 𝐸. (b)If 𝐸 is join- and meet-dense in 𝑋, then all startpoints (resp., endpoints) of 𝑋 are startpoints (resp., endpoints) of 𝐸.

Proof. (a) Suppose that 𝑥 is a startpoint of (𝑋,𝑑). Then our characterization (see Lemma 11) of a startpoint gives 𝑦∈𝑋 such that 𝑥 is a minimal element of 𝑋⧵↓𝑦.
Since 𝑥≰𝑦, by join density of 𝐸 in 𝑋, there must be 𝑒∈𝐸 such that 𝑒≤𝑥 and 𝑒≰𝑦. Hence by the minimality property of 𝑥 we obtain 𝑥=𝑒. Therefore 𝑥 belongs to 𝐸. The dual result is proved analogously.
(b) We continue the proof of part (a) dealing with startpoints. By the additional assumption of meet density of 𝐸 in 𝑋, in the proof of part (a) we can find 𝑦^′∈𝐸 such that 𝑦≤𝑦^′ and 𝑥=𝑒≰𝑦^′. Then 𝑥 is minimal in 𝑋⧵↓𝑦^′. We conclude that 𝑦^′∈𝐸 witnesses that 𝑥∈𝐸 is a startpoint in 𝐸, because 𝐸⊆𝑋. The result on endpoints is proved similarly.

5. Examples

In this section we will discuss various examples in the light of the results in Section 4. Some of the details of the arguments are left to the reader.

Example 21. Let (𝑋,≤) be a partially ordered set with the smallest element 0 and the largest element 1, where 0≠1, equipped with its natural 𝑇₀-quasimetric 𝑑. Given 𝑥∈𝑋, collinearity of (0,1,𝑥) in (𝑋,𝑑) obviously implies that 𝑥=1. However we cannot conclude that 1 is an endpoint of (𝑋,𝑑) with witness 0, since 𝑑(0,1)=0.

Indeed we have the following result.

Proposition 22. Let (𝑋,≤) be a partially ordered set equipped with its natural 𝑇₀-quasimetric 𝑑. A smallest element 0 of 𝑋 cannot be a startpoint of (𝑋,𝑑). Similarly a largest element 1 of 𝑋 is never an endpoint of (𝑋,𝑑).

Proof. Since 0≤𝑥, thus 𝑑(0,𝑥)=0 whenever 𝑥∈𝑋; there cannot be an element in 𝑋 witnessing that 0 is a startpoint in (𝑋,𝑑). The dual statement is proved similarly.

Let (𝑋,≤) be a linearly ordered set, and let 𝑎,𝑏∈𝑋 be such that 𝑎<𝑏, but there does not exist an element 𝑧∈𝑋 such that 𝑎<𝑧<𝑏. As usual, the pair (𝑎,𝑏) is called a jump in 𝑋. Note that for partially ordered sets one also says that𝑏covers 𝑎 in this case; see, for instance, [9, page 11].

Proposition 23. Let (𝑋,≤) be a linearly ordered set equipped with its natural 𝑇₀-quasimetric 𝑑. The first elements of jumps in 𝑋 are exactly the endpoints of (𝑋,𝑑). The second elements of jumps in 𝑋 are exactly the startpoints of (𝑋,𝑑).

Proof. Suppose that (𝑎,𝑏) is a jump in 𝑋. Then 𝑏 is minimal in 𝑋⧵↓𝑎. Hence 𝑏 is a startpoint of (𝑋,𝑑) by Lemma 11.
In order to prove the converse suppose that 𝑏 is a startpoint of (𝑋,𝑑). Then there is 𝑦∈𝑋 such that 𝑏 is minimal in 𝑋⧵↓𝑦. Since ≤ is a linear order, we have that 𝑦<𝑏 and (𝑦,𝑏) is a jump. Similarly one proves the stated result on endpoints.

Corollary 24. In {0,1}, equipped with its usual linear order and natural 𝑇₀-quasimetric, 0 is an endpoint but not a startpoint, while 1 is a startpoint but not an endpoint.
In the set ℤ of integers equipped with the usual linear order and its natural 𝑇₀-quasimetric, each point is an endpoint and a startpoint.
The closed unit interval of the set of rational numbers equipped with its usual linear order and the natural 𝑇₀-quasimetric induced by that order does not have any endpoints nor any startpoints.

Corollary 25. Let (𝑋,≤) be a partially ordered set equipped with its natural 𝑇₀-quasimetric 𝑑. If, for some 𝑐∈𝑋,↓𝑐 consists of two elements, then 𝑐 is a startpoint in (𝑋,𝑑). Similarly, if, for some 𝑐∈𝑋,↑𝑐 consists of two elements, then 𝑐 is an endpoint in (𝑋,𝑑).

Proof. Suppose that ↓𝑐={𝑐,𝑚} with 𝑚≠𝑐. Obviously 𝑐 is a minimal element of 𝑋⧵↓𝑚. The result follows from Lemma 11. The dual result is proved analogously.

Corollary 26. Each atom in a partially ordered set 𝑋 (see [9, page 113] for the definition) with the smallest element and equipped with its natural 𝑇₀-quasimetric 𝑑 is a startpoint of (𝑋,𝑑). Similarly each coatom in a partially ordered set 𝑋 with a largest element and equipped with its natural 𝑇₀-quasimetric 𝑑 is an endpoint of (𝑋,𝑑).

Example 27. For a set 𝑋 with at least one element consider the complete lattice (𝒫(𝑋),⊆) equipped with its natural 𝑇₀-quasimetric 𝑑, where 𝒫(𝑋) is the power set of 𝑋. Then the startpoints of (𝒫(𝑋),𝑑) are exactly the singletons. The endpoints of (𝒫(𝑋),𝑑) are exactly the complements of the singletons.
In fact, for each 𝑥∈𝑋, the startpoint {𝑥} witnesses that 𝑋⧵{𝑥} is an endpoint, and the endpoint 𝑋⧵{𝑥} witnesses that {𝑥} is a startpoint. Observe that for each 𝑥∈𝑋,({𝑥},𝑋⧵{𝑥}) is a completely prime pair.

Example 28. Let ℛ be the usual topology on the set ℝ of the reals equipped with set-theoretic inclusion as a partial order, and let 𝑑 be its natural 𝑇₀-quasimetric. Then there are no startpoints, and exactly the complements of singletons are the endpoints in (ℛ,𝑑).
Let us give a proof of this statement just using the basic definitions. Suppose that 𝐻∈ℛ is a startpoint with witness 𝐻^′∈ℛ. In particular 𝐻⊆𝐻̸^′. Let ℎ∈𝐻⧵𝐻^′. Then we find 𝐻^′′∈ℛ such that ℎ∈𝐻^′′⊂𝐻. It follows that (𝐻^′′,𝐻,𝐻^′) is collinear in (ℛ,𝑑)—a contradiction. We conclude that there are no startpoints in (ℛ,𝑑).
On the other hand fix 𝑥∈ℝ. Then collinearity of (𝑋,𝑋⧵{𝑥},𝐺) with 𝐺∈ℛ implies that 𝐺=𝑋⧵{𝑥}. Thus 𝑋 witnesses that 𝑋⧵{𝑥} is an endpoint in (ℛ,𝑑).
Assume that 𝐻 is an endpoint in (ℛ,𝑑). Then there is a witness 𝐺∈ℛ such that 𝐺⊈𝐻. Let 𝑔∈𝐺⧵𝐻. Then (𝐺,𝐻,𝑋⧵{𝑔}) is collinear in (ℛ,𝑑). By our assumption 𝐻=𝑋⧵{𝑔}. Hence exactly the complements of singletons are the endpoints in (ℛ,𝑑).

Given a nonempty subset 𝐴 of a set 𝑋, by fil{𝐴} we denote the filter generated by the filter base {𝐴} on 𝑋.

Corollary 29. Let 𝒫 be the set of filters (partially ordered under set-theoretic inclusion) on an infinite set 𝑋 and equipped with its natural 𝑇₀-quasimetric 𝑑. Then the set of endpoints of (𝒫,𝑑) consists of all the ultrafilters on 𝑋, and the set of startpoints of (𝒫,𝑑) consists of all the filters 𝑓𝑖𝑙{𝑋⧵{𝑥}} with 𝑥∈𝑋.

Proof. It is well known that each filter on 𝑋 is the intersection of ultrafilters on 𝑋 and that the maximal elements in 𝒫 are the ultrafilters. Hence the endpoints in 𝒫 are exactly the ultrafilters on 𝑋 (see Propositions 15 and 20).
Furthermore 𝒫 has a smallest element, namely, {𝑋}. Since the set of fil{𝐴} with nonempty 𝐴⊆𝑋 is join-dense in 𝒫, the startpoints can only be of the form fil{𝐴} with proper 𝐴⊆𝑋 by Propositions 20 and 22. We know by Corollary 26 that, for all 𝑥∈𝑋,fil{𝑋⧵{𝑥}} is a startpoint in 𝒫. We finally show that there are no other startpoints in (𝒫,𝑑).
Let 𝐴⊆𝑋 be such that 𝑋⧵𝐴 contains at least two points. Take any 𝒢∈𝒫 such that fil{𝐴}⊆𝒢̸. We consider two cases.
Case  1. (∩{𝐺∶𝐺∈𝒢})⧵𝐴=∅. Then choose 𝑎^′∈𝑋⧵𝐴. It follows that fil{𝐴} is not minimal in 𝒫⧵↓𝒢, since fil{𝐴∪{𝑎^′}}⊂fil{𝐴} and also fil{𝐴∪{𝑎^′}}⊈𝒢. Indeed if 𝐴∪{𝑎^′}∈𝒢, then there is 𝑌^′∈𝒢 such that 𝑎^′∉𝑌^′; hence (𝐴∪{𝑎^′})∩𝑌^′⊆𝐴 and then fil{𝐴}⊆𝒢—a contradiction.
Case  2. There is 𝑎^′∈(∩{𝐺∶𝐺∈𝒢})⧵𝐴. By our assumption about 𝐴 we can choose 𝑎^′′∈𝑋⧵(𝐴∪{𝑎^′}). Then fil{𝐴∪{𝑎^′′}}⊂fil{𝐴} and also fil{𝐴∪{𝑎^′′}}⊆𝒢̸, because 𝐴∪{𝑎^′′}∉𝒢, since 𝑎^′∉𝐴∪{𝑎^′′}. It follows that fil{𝐴} is not minimal in 𝒫⧵↓𝒢.
Thus we are done, since no 𝒢∈𝒫 exists that could witness that fil{𝐴} is a startpoint of (𝒫,𝑑).

6. The Dedekind-MacNeille Completion versus the 𝑞-Hyperconvex Hull

Some of the results mentioned in the previous sections may have reminded the reader of the theory of the Dedekind-MacNeille completion (cf. also [16]). Of course this is not accidental but can be explained categorically (see, e.g., [5, 8]).

For the following discussion we need some basic facts from the theory of the Dedekind-MacNeille completion of a partially ordered set (see, e.g., [9, page 166]).

Let (𝑋,≤) be a partially ordered set, and let 𝐴⊆𝑋. Then we define the set of upper bounds of 𝐴, that is, 𝐴^𝑢={𝑥∈𝑋∶𝑎≤𝑥 whenever 𝑎∈𝐴} and the set of lower bounds of 𝐴, that is, 𝐴^ℓ={𝑥∈𝑋∶𝑎≥𝑥 whenever 𝑎∈𝐴}. Let DM(𝑋)={𝐴⊆𝑋∶𝐴^𝑢ℓ=𝐴}. The partially ordered set (DM(𝑋),⊆) is a complete lattice. It is known as the Dedekind-MacNeille completion of 𝑋. Furthermore 𝜙∶𝑋→DM(𝑋) defined by 𝜙(𝑥)=↓𝑥 is an order embedding such that 𝜙(𝑋) is both join-dense and meet-dense in DM(𝑋). This is indeed the characteristic property of the Dedekind-MacNeille completion (cf. [9, Theorem 7.41]).

Proposition 30. Let (𝑋,≤) be a partially ordered set and 𝑑 its natural 𝑇₀-quasimetric. Furthermore let 𝐷 be the natural 𝑇₀-quasimetric of (𝐷𝑀(𝑋),⊆). Then (𝑋,𝑑) and (𝐷𝑀(𝑋),𝐷) have the same startpoints (resp., endpoints).

Proof. For the proof we consider 𝑋 a subset of DM(𝑋). By Proposition 20(b) each startpoint (resp., endpoint) of (DM(𝑋),𝐷) is a startpoint (resp., endpoint) of (𝑋,𝑑), since 𝑋 is both join-dense and meet-dense in DM(𝑋). Suppose now that 𝑥 is a startpoint of (𝑋,𝑑) with witness 𝑦∈𝑋. Let 𝑓∈DM(𝑋) be such that (𝑓,𝑥,𝑦) is collinear in (DM(𝑋),𝐷). Thus 𝑥≰𝑦, and, therefore, ↓𝑥⊈↓𝑦,𝑓⊆↓𝑥, and 𝑓⊈↓𝑦. Since 𝑋, is join-dense in DM(𝑋), there is 𝑓^′∈𝑋 such that ↓𝑓^′⊆𝑓 and ↓𝑓^′⊈↓𝑦. Thus ↓𝑓^′⊆𝑓⊆↓𝑥 and (𝑓^′,𝑥,𝑦) is collinear in (𝑋,𝑑). Hence 𝑓^′=𝑥 and therefore 𝑓=↓𝑥, too. Consequently 𝑥 is a startpoint with witness 𝑦 in (DM(𝑋),𝐷). The dual result is proved analogously.

Example 31. We continue the discussion of Example 18 (see [9, page 169] and Figure 2).
Considering 𝑃₄ as a subset of DM(𝑃₄), in the light of Proposition 30 and according to Corollary 17 in the complete lattice DM(𝑃₄), the set of the startpoints of 𝑃₄ becomes the set of the (completely) join-irreducible elements of DM(𝑃₄), and the set of the endpoints of 𝑃₄ becomes the set of the (completely) meet-irreducible elements of DM(𝑃₄).

Figure 2 

Hasse diagram of DM(𝑃4).

We will say that a partially ordered set (𝑋,≤) with its natural 𝑇₀-quasimetric 𝑑 is 𝑜-hyperconvex if, for any 𝐴⊆𝑋 and any families (𝑟_𝑥)_𝑥∈𝐴,(𝑠_𝑥)_𝑥∈𝐴 in {0,1} satisfying that 𝑑(𝑥,𝑦)≤𝑟_𝑥+𝑠_𝑦 whenever 𝑥,𝑦∈𝐴, it follows that ⋂_𝑥∈𝐴(𝐶_𝑑(𝑥,𝑟_𝑥)∩𝐶_𝑑⁻¹(𝑥,𝑠_𝑥))≠∅. Observe that (𝑋,𝑑) is 𝑜-hyperconvex provided that (𝑋,𝑑) is 𝑞-hyperconvex.

Proposition 32. Let (𝑋,≤) be a complete lattice and 𝑑 its natural 𝑇₀-quasimetric on 𝑋. Then (𝑋,𝑑) is 𝑜-hyperconvex.

Proof. Let 𝐴⊆𝑋, and let (𝑟_𝑥)_𝑥∈𝐴 and (𝑠_𝑥)_𝑥∈𝐴 be families in {0,1} such that 𝑑(𝑥,𝑦)≤𝑟_𝑥+𝑠_𝑦 whenever 𝑥,𝑦∈𝐴. Set 𝑅={𝑥∈𝐴∶r_𝑥=0} and 𝑆={𝑥∈𝐴∶𝑠_𝑥=0}. Note that 𝑅∩𝑆 contains at most one element, since (𝑋,𝑑) is a 𝑇₀-space. We observe that 𝑥,𝑦∈𝑅∩𝑆 implies that 𝑑(𝑥,𝑦)≤𝑟_𝑥+𝑠_𝑦 and 𝑑(𝑦,𝑥)≤𝑟_𝑦+𝑠_𝑥; thus 𝑑(𝑥,𝑦)=0=𝑑(𝑦,𝑥), and hence 𝑥=𝑦.
Furthermore by our assumption, obviously we have that 𝑅⊆𝑆^ℓ, since 𝑑(𝑟,𝑠)≤0+0 whenever 𝑟∈𝑅 and 𝑠∈𝑆. Since 𝑋 is a complete lattice, ∨𝑅 exists.
Then ∨𝑅∈_𝑥∈𝑅↑𝑥∩_𝑥∈𝑆=↓𝑥_𝑥∈𝐴𝐶_𝑑𝑥,𝑟_𝑥∩_𝑥∈𝐴𝐶_𝑑⁻¹𝑥,𝑠_𝑥,(7) since 𝐶_𝑑(𝑥,𝑟_𝑥)=𝑋 if 𝑥∈𝐴⧵𝑅 and 𝐶_𝑑⁻¹(𝑥,𝑠_𝑥)=𝑋 whenever 𝑥∈𝐴⧵𝑆. Thus (𝑋,𝑑) is 𝑜-hyperconvex.

As an illustration it may be useful to include here the following simple example (see [4, Example 8]).

Example 33. Let 𝑋={0,1} be equipped with its usual order ≤ and with its natural 𝑇₀-quasimetric 𝑑. Then (𝑄_𝑋,𝐷) can be identified with ([0,1],𝑢) under the obvious inclusion 𝑋→[0,1]. Hence (𝑋,𝑑) is not 𝑞-hyperconvex, although (𝑋,≤) is a complete lattice.

For the following result compare the discussion preceding [8, Lemma 2.5].

Proposition 34. Let (𝑋,𝑑) be a bounded 𝑞-hyperconvex 𝑇₀-quasimetric space and ≤ its specialization order. Then (𝑋,≤) is a complete lattice.

Proof. Suppose that 𝑔∈[0,∞) is an upper bound of 𝑑.
Let 𝐴⊆𝑋 and 𝑥∈𝑋. Set 𝑟_𝑥=0 if 𝑥∈𝐴, and 𝑟_𝑥=𝑔 otherwise. Furthermore let 𝑠_𝑥=0 if 𝑥∈𝐴^𝑢, and 𝑠_𝑥=𝑔 otherwise.
Consider now arbitrary 𝑥,𝑦∈𝑋. Assume first that 𝑥∈𝐴 and 𝑦∈𝐴^𝑢. Then 𝑥≤𝑦 and 𝑑(𝑥,𝑦)=0=𝑟_𝑥+𝑠_𝑦.
Suppose now that 𝑥∉𝐴 or 𝑦∉𝐴^𝑢. Consequently 𝑟_𝑥+𝑠_𝑦≥𝑔≥𝑑(𝑥,𝑦). We have shown that for ((𝐶_𝑑(𝑥,𝑟_𝑥))_𝑥∈𝑋;(𝐶_𝑑⁻¹(𝑥,𝑠_𝑥))_𝑥∈𝑋) the hypothesis of the condition of 𝑞-hyperconvexity is satisfied.
We conclude that there is 𝑦∈_𝑥∈𝑋𝐶_𝑑𝑥,𝑟_𝑥∩_𝑥∈𝑋𝐶_𝑑⁻¹𝑥,𝑠_𝑥⊆_𝑎∈𝐴↑𝑎∩_{𝑏∈𝐴𝑢}↓𝑏.(8)
Consequently 𝑦=∨𝐴. (Of course, similarly one could show that ∧𝐴 exists.) We deduce that (𝑋,≤) is a complete lattice.

Example 35. Recall that (ℝ,𝑢) is a 𝑞-hyperconvex 𝑇₀-quasimetric space (see [4, Example 1]). The specialization order ≤ of that space is the standard order on ℝ; hence (ℝ,≤) is not a complete lattice. So boundedness cannot be omitted in Proposition 34.

Remark 36. Let (𝑋,≤) be a partially ordered set and 𝑑 its natural 𝑇₀-quasimetric. If (𝑋,𝑑) is 𝑜-hyperconvex, then (𝑋,≤) is a complete lattice. This is a consequence of the proof of Proposition 34 by setting 𝑔=1, since we have ≤_𝑑=≤.

Indeed, more generally, our next result shows explicitly how, given a partially ordered set (𝑋,≤) equipped with its natural 𝑇₀-quasimetric 𝑑, the 𝑞-hyperconvex hull (𝑄_(𝑋,𝑑),𝐷) of (𝑋,𝑑) contains the Dedekind-MacNeille completion of 𝑋.

Lemma 37. Let (𝑋,≤) be a partially ordered set and 𝑑 its natural 𝑇₀-quasimetric. Furthermore let 𝐹_𝑋 be the set of all those minimal ample function pairs (𝑓₁,𝑓₂) on (𝑋,𝑑) that only attain the values 0 and 1.
Consider an arbitrary pair (𝑓₁,𝑓₂) of functions 𝑋→{0,1}. Then the following conditions are equivalent.(a)(𝑓₁,𝑓₂)∈𝐹_𝑋. (b)𝑓₁⁻¹{0}=(𝑓₂⁻¹{0})^𝑢 and 𝑓₂⁻¹{0}=(𝑓₁⁻¹{0})^ℓ.(c)(𝑓₂⁻¹{0})^𝑢ℓ=𝑓₂⁻¹{0} and 𝑓₁(𝑥)=sup_𝑦∈𝑋(𝑑(𝑦,𝑥)−̇𝑓₂(𝑦)) whenever 𝑥∈𝑋.

Proof. (a)→(b)∶ given 𝑥∈𝑋, consider 𝑓₁(𝑥)=sup_𝑦∈𝑋𝑑𝑦,𝑥−̇𝑓₂𝑦(9) (see (∗) from Section 2).
Since the functions 𝑓₁,𝑓₂ and 𝑑 attain only the values 0 and 1, we see that these equations are equivalent to 𝑓₁⁻¹{0}=(𝑓₂⁻¹{0})^𝑢. Indeed, given 𝑥∈𝑋, we have that 𝑓₁(𝑥)=0 if and only if, for any 𝑦∈𝑋,𝑑(𝑦,𝑥)=1 implies that 𝑓₂(𝑦)=1, and if and only if, for any 𝑦∈𝑋,𝑓₂(𝑦)=0 implies that 𝑑(𝑦,𝑥)=0 if and only if 𝑥∈(𝑓₂⁻¹{0})^𝑢.
Similarly one verifies that 𝑓₂⁻¹{0}=(𝑓₁⁻¹{0})^ℓ is equivalent to 𝑓₂(𝑥)=sup_𝑦∈𝑋(𝑑(𝑥,𝑦)−̇𝑓₁(𝑦)) whenever 𝑥∈𝑋.
Since (𝑓₁,𝑓₂)∈𝐹_𝑋⊆𝑄_𝑋, condition (b) is satisfied.
(b)→(c)∶ by (b) we conclude that (𝑓₂⁻¹{0})^𝑢ℓ=𝑓₂⁻¹{0}. Furthermore the second part of (c) is equivalent to 𝑓₁⁻¹{0}=(𝑓₂⁻¹{0})^𝑢, as we have just noted above. Hence condition (c) is satisfied.
(c)→(a): observe that (𝑓₂⁻¹{0})^𝑢ℓ=𝑓₂⁻¹{0} and 𝑓₁⁻¹{0}=(𝑓₂⁻¹{0})^𝑢 together imply that 𝑓₂⁻¹{0}=(𝑓₁⁻¹{0})^ℓ. But the latter equality is equivalent to 𝑓₂(𝑥)=sup_𝑦∈𝑋(𝑑(𝑥,𝑦)−̇𝑓₁(𝑦)) whenever 𝑥∈𝑋, as we have observed above. Thus (𝑓₁,𝑓₂)∈𝐹_𝑋, and condition (a) holds.

Proposition 38. Let (𝑋,≤) be a partially ordered set with its natural 𝑇₀-quasimetric 𝑑, and let 𝐹_𝑋 be defined as in Lemma 37.
Then the map 𝜓∶(𝐹_𝑋,≤_𝐷)→(𝐷𝑀(𝑋),⊆) defined by (𝑓₁,𝑓₂)↦𝑓₂⁻¹{0} is an order isomorphism between 𝐹_𝑋 (equipped with the specialization order ≤_𝐷 induced on 𝐹_𝑋 by the 𝑇₀-quasimetric 𝐷 of the 𝑞-hyperconvex hull of (𝑋,𝑑)) and the Dedekind-MacNeille completion (𝐷𝑀(𝑋),⊆) of 𝑋. Furthermore for each 𝑥∈𝑋,𝜓(𝑓_𝑥)=↓𝑥.

Proof. By Lemma 37 each set 𝑓₂⁻¹{0} belongs to the Dedekind-MacNeille completion, since 𝑓∈𝐹_𝑋. Moreover for each 𝑥∈𝑋, 𝑓_𝑥∈𝐹_𝑋. Also for each 𝑥∈𝑋, obviously (𝑓_𝑥)₂⁻¹{0}=↓𝑥. The specialization order ≤_𝐷 induced on 𝐹_𝑋 by the 𝑇₀-quasimetric 𝐷 is defined by 𝑓⪯𝑔 if and only if 𝐷(𝑓,𝑔)=0 whenever 𝑓,𝑔∈𝐹_𝑋. Hence we have for any 𝑓,𝑔∈𝐹_𝑋, 𝐷𝑓,𝑔=sup_𝑥∈𝑋𝑔₂(𝑥)−̇𝑓₂(𝑥)=0(10) if and only if 𝜓𝑓=𝑓₂⁻¹{0}⊆𝑔₂⁻¹{0}=𝜓𝑔.(11)
(Of course this means exactly that 𝑔₂≤𝑓₂ with respect to the usual pointwise order on real-valued functions.) In particular 𝜓 is injective. Furthermore for 𝐴⊆𝑋 with 𝐴^𝑢ℓ=𝐴 we define 𝑓₂∶𝑋→{0,1} so that 𝐴=𝑓₂⁻¹{0} and 𝑓₁(𝑥)=sup_𝑦∈𝑋(𝑑(𝑦,𝑥)−̇𝑓₂(𝑦)) whenever 𝑥∈𝑋. Then 𝑓=(𝑓₁,𝑓₂)∈𝐹_𝑋 according to Lemma 37 and 𝜓(𝑓)=𝐴, and hence 𝜓 is surjective.
We conclude that the set 𝐹_𝑋 can be identified with the ground set of the Dedekind-MacNeille completion of 𝑋, and 𝜓∶(𝐹_𝑋,≤_𝐷)→(𝐷𝑀(𝑋),⊆) is an order isomorphism satisfying 𝜓(𝑓_𝑥)=↓𝑥.

Remark 39. Given a partially ordered set (𝑋,≤) equipped with its natural 𝑇₀-quasimetric 𝑑 and its 𝑞-hyperconvex hull 𝑄_(𝑋,𝑑), the subspace 𝑆 identified above with 𝐷𝑀(𝑋) in 𝑄_(𝑋,𝑑) is obviously characterized by the property that it is the largest subspace of 𝑄_(𝑋,𝑑) containing 𝑒(𝑋) and such that the 𝑇₀-quasimetric 𝐷 restricted to 𝑆×𝑆 attains only values in {0,1}.

Example 40. Let 𝑋={0,1} be equipped with the discrete order = . As we have observed above, the natural 𝑇₀-quasimetric on 𝑋 is the discrete metric. Furthermore (𝑄_𝑋,𝐷) can be identified with the set 𝑌=[0,1]×[0,1] equipped with the 𝑇₀-quasimetric 𝐷𝛼₁,𝛼₂,𝛽₁,𝛽₂=𝛼₁−̇𝛽₁∨𝛼₂−̇𝛽₂(12) whenever (𝛼₁,𝛼₂),(𝛽₁,𝛽₂)∈𝑌, where 0 is identified with (0,1) and 1 is identified with (1,0) (see [6, Example 4]). Of course the Dedekind-MacNeille completion of (𝑋,𝑑) consists only of the four corner points of 𝑌 endowed with the induced specialization order on 𝑌 (see Example 𝑃₂ in [9, page 169]).

Problem 1. Given a 𝑇₀-quasimetric space (𝑋,𝑑), compare the set of the endpoints of (𝑋,𝑑) with the set of the endpoints of (𝑄_𝑋,𝐷).

In this paper we established that these two sets are equal if 𝜏(𝑑^𝑠) is compact.

7. Extended 𝑇₀-Quasimetrics

Given a partially ordered set (𝑋,≤), in the last three sections we worked with the natural 𝑇₀-quasimetric 𝑑 of 𝑋. The choice of the value 1 in the definition of the natural 𝑇₀-quasimetric looked rather arbitrary. A more canonical approach can be given if one uses extended 𝑇₀-quasimetrics.

Remark 41. Let (𝑋,≤) be a partially ordered set. For each 𝑥,𝑦∈𝑋, we set 𝜌(𝑥,𝑦)=0 if 𝑥≤𝑦 and 𝜌(𝑥,𝑦)=∞ otherwise (see, e.g., [8, Section 2.1]). Obviously, 𝜌 is an extended 𝑇₀-quasimetric on 𝑋.
We note however that, for any 𝑎∈[0,∞], we have ∞+𝑎=∞, but 1+1=2 (so that {0,1} is not closed under the operation of addition +). Therefore we obtain distinct theories of collinearity by considering (𝑋,𝑑) or (𝑋,𝜌), respectively. For our purposes the 𝑇₀-quasimetric 𝑑 seems to lead to a more interesting theory. We finally show how our characterizations of endpoints and startpoints (see Lemma 11 and Corollary 12) have to be modified to be applied to our new setting.

Proposition 42. Let (𝑋,≤) be a partially ordered set, 𝜌 its natural extended 𝑇₀-quasimetric, and 𝑥,𝑦∈𝑋. Then 𝑥 is an endpoint with witness 𝑦 in (𝑋,𝜌) if and only if one has {𝑥}=𝑋⧵↑𝑦.
Furthermore 𝑥 is a startpoint with witness 𝑦 in (𝑋,𝜌) if and only if one has {𝑥}=𝑋⧵↓𝑦.

Proof. Suppose that 𝑥 is an endpoint with witness 𝑦 in (𝑋,𝜌). Therefore 𝜌(𝑦,𝑥)=∞, and, for all 𝑎∈𝑋,𝜌(𝑦,𝑥)+𝜌(𝑥,𝑎)=𝜌(𝑦,𝑎) implies that 𝑥=𝑎. Hence 𝑥∈𝑋⧵↑𝑦. Consider any 𝑎∈𝑋⧵↑𝑦. Thus ∞=𝜌(𝑦,𝑥)+𝜌(𝑥,𝑎)=𝜌(𝑦,𝑎), which by our assumption implies that 𝑥=𝑎. Therefore {𝑥}=𝑋⧵↑𝑦.
For the converse suppose that {𝑥}=𝑋⧵↑𝑦. Then 𝜌(𝑦,𝑥)=∞. Let 𝑎∈𝑋 be such that 𝜌(𝑦,𝑥)+𝜌(𝑥,𝑎)=𝜌(𝑦,𝑎). Then 𝜌(𝑦,𝑎)=∞ and 𝑎∈𝑋⧵↑𝑦={𝑥}. Consequently 𝑎=𝑥. Thus 𝑥 is an endpoint with witness 𝑦 in (𝑋,𝜌). The dual result is proved analogously.

Example 43. Let us consider the set ℤ of the integers equipped with its usual linear order, and let 𝑑 be its natural 𝑇₀-quasimetric and, 𝜌 its natural extended 𝑇₀-quasimetric respectively. Then (ℤ,𝜌) does not have any endpoints nor any startpoints by Proposition 42, while in (ℤ,𝑑) each point is an endpoint and a startpoint (see Corollary 24).

8. Conclusion

We showed that for any join compact 𝑇₀-quasimetric space (𝑋,𝑑) the set of endpoints (resp., startpoints) of (𝑋,d) is equal to the set of endpoints (resp., startpoints) of its 𝑞-hyperconvex hull (𝑄_𝑋,𝐷). We also specialized some of our earlier results on endpoints contained in [6] to two-valued 𝑇₀-quasimetric spaces. In particular we observed that in the case of a complete lattice 𝑋 and its natural 𝑇₀-quasimetric 𝑑 the startpoints (resp., endpoints) of (𝑋,𝑑) are exactly the completely join-irreducible (resp., the completely meet-irreducible) elements. For a partially ordered set we finally explored the connection between its Dedekind-MacNeille completion and the 𝑞-hyperconvex hull of its natural 𝑇₀-quasimetric space.

Acknowledgments

The authors would like to thank the National Research Foundation of South Africa for partial financial support. This research was also supported by a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme. Finally the authors would like to thank the referee for his or her suggestions.