Quasipseudometric spaces and topologicalproperties

Cynthia Basileu, Michel LamureUniversity Lyon 1

FranceEmail: {cbasileu, lamure}@univ-lyon1.fr

Soufian Ben Amor, Marc BuiEcole Pratique des Hautes Etudes

FranceEmail: {sofiane.benamor, marc.bui}@ephe.sorbonne.fr

Abstract—In this paper, we introduce a new concept of metric:quasipseudometric, by relaxing the usual axiomatic of metrics.We study the consequences from a theoretical point of view. Inparticular, we focus ourselves on the ways to associate generaltopological structures to such a quasipseudometric. We show weget a weakened version of a topology: a pretopology, which canhowever be endowed with interesting properties.

I. INTRODUCTION

Clustering techniques are based, as a rule, on the conceptof distance that supposes a precise axiomatic. This axiomaticsometimes proves to be very coercive, in particular in theapplications in social sciences. In various cases, a problemthat arises is the property of symmetry which expresses thatthe distance from x to y is equal to the distance from y to x.The question is therefore to know what it happens when thishypothesis is relaxed.

Obviously, the first questions, in this case, are relevant oftheoretical points:

• Is it possible to define a concept of ”metric” without thesymmetry axiom, but with sufficient properties to build acoherent theory ?

• What are the basic properties of such a ”metric” ?• What kind of topological space can we endow the metric

space with ?

In this work, we propose to give some answers thesequestions related to such a measure of distance fulfillingneither the property of symmetry nor the triangular inequalityproperty. We derive from this definition the first propertiesrelative to the structures which the set could be endowed with.In particular, we show that it is not more possible to get atopology and that the structures that we can drift of such adistance measure are only pretopological structures ([6] [7][8] [2]), however endowed with interesting properties ([5]).This leads us to come back to the early history of generaltopology ([4],[3],[9],[1]).

II. DEFINITIONS AND BASIC PROPERTIES

In this section, we shall focus ourselves on defining moregeneral spaces than metric ones and we shall study propertiesof those spaces. This extension of metric spaces is obtained byway of keeping only the first axiom of a metric. We presentall definitions and basic results about quasipseudometricshereafter.

Definition 1: (Quasipseudometric) Let be E a non emptyset, we call quasipseudometric on E, any mapping from E×Einto P (E) such as ∀(x, y), (x, y) ∈ E×E, d(x, y) = 0 ⇔ x =y

Example 1: Let E = R2.

For any x = (x1, x2) and y = (y1, y2) in E, we set:d(x, y) = 2(x1 − y1) + 2(x2 − y2) if y1 ≤ x1 and y2 ≤ x2

d(x, y) = (y1 − x1) + (y2 − x2) if y1 ≥ x1 and y2 ≥ x2

d(x, y) = 2(x1 − y1) + (y2 − x2) if y1 < x1 and x2 < y2

d(x, y) = 2(x2 − y2) + (y1 − x1) if y1 > x1 and x2 < y2

It is obvious to see that d(x, y) = 0 ⇔ x = y and furthermore,if we take x = (0, 0) and y = (0, 1) we get d(x, y) = 1 andd(y, x) = 2. Thus, d is a quasipseudometric on E.

Example 2: Let E be the set E = {x, y, z, t} and � abinary relationship on E characterized by the following table:

x y z tx 0 1 0 1y 0 0 1 1z 1 1 0 0t 0 0 1 0

We define d as:

d : E × E → R+

∀a ∈ E, d(a, a) = 0∀a ∈ E,∀b ∈ E, d(a, b) = n(a, b) where n(a, b) is the

length of the shortest path from a to b. So, we get thefollowing distance table:

d x y z tx 0 1 2 1y 2 0 1 1z 1 1 0 2t 2 2 1 0

It is not a symmetric table, which shows that d is a quasipseu-dometric, not a metric in the usual sense.

Example 3: Let us consider E = {a, b, c, d, e} and adistance table on E given by:

978-1-4244-8075-3/10/$26.00 ©2010 IEEE

d a b c d ea 0 1

√2

√17 4

b 1 0 1 4√

17c

√2 1 0 3

√10

d√

17 4 3 0 1e 4

√17

√10 1 0

d is a classical metric on E. Let us now consider δ definedon E by: δ(x, y) = k ⇔ y is the kth nearest neighbor of x.Then, we get the following table for δ:

δ a b c d ea 0 1 2 4 3b 1.5 0 1.5 3 4c 2 1 0 3 4d 4 3 2 0 1e 3 4 2 1 0

We can see that δ fulfills axioms of a quasipseudometric, notof a metric.

Definition 2: (Quasipseudometric space) The couple (E, d)where E is a non empty set and d is a quasipseudometric onE, is called quasipseudometric space

In clustering techniques, we are used to work with metricswhich do not fulfill the third axiom, the triangle axiom.We know it is without consequences on existing methodsof clustering. However, when the axiom of symmetry is notfulfilled, we cannot use the classical methods of clusteringand we have to design new ones. We also can imagine anintermediate case between the case where d is symmetric andthe case where d is not symmetric.

Definition 3: Let (E, d) be a quasipseudometric space.If there exists a mapping t from E into itself such that∀(x, y), (x, y) ∈ E2, d(x, y) = d(t(y), t(x)), then we say that(E, d) is t-pseudosymmetric

We can note that, if t is identity mapping, we get the symmetryproperty. From now, we shall work with a quasipseudometricd which is not a symmetric one.

Definition 4: Let (E, d) a quasipseudometric space, we callsurface of symmetry the set S defined by S = {(x, y ∈E2|d(x, y) = d(y, x)}Then, it is obvious to note that:

Property 1: Let (E, d) a quasipseudometric space with asurface of symmetry S

• S = ∅• ∀(x, y) ∈ E2, (x, y) ∈ S ⇔ (y, x) ∈ S

Moreover, we can note that S is a neighborhood of thediagonal Δ of E (Δ = (x, x) | x ∈ E). It is also obvious thatd is a symmetric quasipseudometric if and only if S = E×E.By analogy with metric spaces, we can study the concept ofopen or closed ball, with center x and radius r, but in the caseof a quasipseudometric space, we have to distinguish betweenright and left balls.

Definition 5: Let (E, d) be a quasipseudometric space, r apositive real number and x a point of E

• We call half open right ball with center x and radius r,the set, noted Bd(x, r), defined by

Bd = {y ∈ E|d(x, y) < r}• We call half closed right ball with center x and radius r,

the set, noted Bd(x, r), defined by

Bd = {y ∈ E|d(x, y) ≤ r}• We call half open left ball with center x and radius r, the

set, noted Bg(x, r), defined by

Bg = {y ∈ E|d(y, x) < r}• We call half closed left ball with center x and radius r,

the set, noted Bg(x, r), defined by

Bg = {y ∈ E|d(y, x) ≤ r}

Example 4: If we consider the previous example 1 with x =(0, 0) and r = 1, we get the following half right and left balls(see Figures 1, 2):

Fig. 1. Half right ball.

Fig. 2. Half left ball.

Definition 6: Let us consider a quasipseudometric space(E, d). For any x in E and any r positive real number:

• We call lower open ball (resp. lower closed ball), withcenter x and radius r, the set, noted Binf (x, r) (resp.Binf (x, r)), defined by Binf (x, r) = Bd(x, r)∩Bg(x, r)(resp. Binf (x, r) = Bd(x, r) ∩ Bg(x, r)).

• We call upper open ball (resp. lower closed ball), withcenter x and radius r, the set, noted Bsup(x, r) (resp.Bsup(x, r)), defined by Bsup(x, r) = Bd(x, r)∪Bg(x, r)(resp. Bsup(x, r) = Bd(x, r) ∪ Bg(x, r))

We then get the obvious following result:

Property 2: Let (E, d) be a quasipseudometric space.(i) ∀r, r > 0,∀x, x ∈ E,∀y, y ∈ E, y ∈ Bd(x, r) ⇔ x ∈Bg(y, r)(ii) ∀r, r > 0,∀x, x ∈ E,∀y, y ∈ E, y ∈ Binf (x, r) ⇔ x ∈Binf (y, r)(iii) ∀r, r > 0,∀x, x ∈ E,∀y, y ∈ E, y ∈ Bsup(x, r) ⇔ x ∈Bsup(y, r)(iv) ∀r, r > 0,∀x, x ∈ E, x ∈ Bd(x, r) ∩ Bg(x, r)That last proposition obviously holds for closed balls.Let us suppose that the quasipseudometric d is t-pseudosymmetric, then ∀(x, y) ∈ E2, d(x, y) = d(t(y), t(x)). Soy ∈ Bd(x, r) ⇔ d(x, y) < r⇔d(t(y), t(x)) < r⇔t(y) ∈ Bg(x, r).If t−1(A) denotes the set defined by t−1(A) = {x, x ∈E|t(x) ∈ A}, we can write:

Property 3: Let (E, d) be a quasipseudometric space, if dis t-pseudo symmetric, then :

(i) Bd(x, r) = t−1(Bg(t(x), r))(ii) Bd(x, r) = t−1(Bg(t(x), r))

III. QUASIPSEUDOMETRICS AND PRETOPOLOGICAL

SPACES

In this section, we shall study how to build pretopologicalstructures related to a given quasipseudometric on a set E,then we will study what are their properties.

First, we shall consider a given threshold r, r > 0.Let us consider Br(x) = {Bd(x, r), Bg(x, r)} and the prefilterVr of subsets of E generated by Br(x), i.e. Vr = {V, V ∈P(E) | V ⊃ Bd(x, r) or V ⊃ Bg(x, r)}. Then, let us definethe mapping ar from P(E) into itself by

ar(A) = {x, x ∈ E|∀V, V ∈ Vr, V ∩ A = ∅}Definition 7: The pretopological structure Pr defined onto

E by families Vr is called the r-pretopology associated to d.

This pretopology is a V-type one, but generally speaking itis not a VD-type one (see [2]). If we consider the first exampleof the previous section, in the case where r = 1, B1(0) ={Bd(0, 1), Bg(0, 1)}. Then V ∈ V1(0) ⇔ V ⊃ Bd(0, 1) orV ⊃ Bd(0, 1). Let us consider the case illustrated by thefigure 3:

Fig. 3. Neighbourhood.

V is a neighbourhood of 0, W is also a neighbourhood of0, but V ∩ W is not a neighbourhood of 0. This implies thatV1(0) is not a filter of subsets of E and then the pretopologyis not a VD one.

The pseudoclosure map can then be expressed as follows:

Property 4: ∀A,A ∈ P(E)ar(A) = {x ∈ E | Bd(x, r) ∩ A = ∅ and Bg(x, r) ∩ A = ∅}Thus, we know the main properties of the pretopology Pr witha fixed r.

Let us examine what it can be said when we consider twodifferent pretopologies generated by two distinct thresholds r1

and r2. Let us consider the case where r1 ≤ r2. Then, it isimmediate to note that :∀x, x ∈ E, Bd(x, r1) ⊂ Bd(x, r2)∀x, x ∈ E, Bg(x, r1) ⊂ Bg(x, r2). Let us note by Br1(x) andBr2(x) the corresponding neighborhoods basis.

∀V, V ∈ Vr2(x), V ⊃ Bd(x, r2) ∨ V ⊃ Bd(x, r2)

⇒ ∀V, V ∈ Vr2(x), V ⊃ Bd(x, r1) ∨ V ⊃ Bd(x, r1)

⇒ ∀V, V ∈ Vr2(x) ⇒ V ∈ Vr1(x)

Then:

Property 5: If r1 ≤ r2, the pretopology Pr1 is coarser thanthe pretopology Pr2 (we denote Pr1 ≺ Pr2)

These pretopologies, generated by a given threshold, rely uponthe concepts of half right balls and half left balls. But, we alsohave the concepts of sup and inf balls.It is possible to generatetwo new pretopologies from them by setting:

Pr = {V(x), x ∈ E}where

V(x) = {V ⊂ E/V ⊃ Binf (x, r)}Pr = {V(x), x ∈ E}

whereV(x) = {V ⊂ E/V ⊃ Bsup(x, r)}

These two pretopologies are obviously VD pretopologies andby the definitions of Binf (x, r) and Bsup(x, r), we get:

Property 6: The pretopologies Pr and Pr are VD pre-topologies and we have Pr ≺ Pr ≺ Pr

Up to now, we worked with pretopologies generated by givinga threshold r. The selection of that threshold may introduce abias, so it would be important to be able to define associatedpretopologies to a quasipseudometric without considering sucha threshold. For that, let us consider

∀x, x ∈ E,B(x) = {Bd(x, r), Bg(x, r), r > 0}and let denote by V(x) the prefilter generated by B(x), i.e.V ∈ V(x)⇔(∃r, r > 0, V ⊃ Bd(x, r)) ∨ (∃r′, r′ > 0, V ⊃ Bg(x, r′)) Sowe can put

∀A,A ⊂ E, a(A) = {x ∈ E/∀V, V ∈ V(x), V ∩ A = ∅}It is obvious to see that :a(∅) = ∅∀A,A ∈ E,A ⊂ a(A)It follows the definition of the induced pretopology on E bythe quasipseudometric :

Definition 8: (Induced pretopology) The pretopology whichis defined by the family V(x)given above and the pseudoclosure a of which is the function defined above, is calledthe pretopology induced on E by the quasipseudometric d

Property 7: ∀A,A ⊂ Ea(A) = {x ∈ E | ∀r, r > 0, (Bd(x, r)∩A = ∅)∧ (Bg(x, r)∩A = ∅)}Proof.

∀A, a(A) = {x ∈ E/∀V, V ∈ V(x), V ∩ A = ∅}⇒x ∈ A ⇒ ∀r, r > 0, (Bd(x, r) ∩A = ∅) ∧ (Bg(x, r) ∩A = ∅)Conversely, let us suppose that : ∀r, r > 0Bd(x, r)∩A = ∅∧Bg(x, r)∩A = ∅)∨(∃V 0, V 0 ∈ V(x), V 0∩A = ∅)As ∃r0 such that V 0 ⊃ Bd(x, r0) or V 0 ⊃ Bg(x, r0)This leads us to Bd(x, r0) ∩ A = ∅ or Bg(x, r0) ∩ A = ∅which is contradictory.Q.E.D.

Property 8: The pretopology induced by a quasipseudomet-ric d is a V one.

Proof.let us suppose that A ⊂ Bx ∈ a(A) ⇔ ∀r, r > 0, Bd(x, r) ∩ A = ∅ ∧ Bg(x, r) ∩ A = ∅⇒

x ∈ a(A) ⇔ ∀r, r > 0, Bd(x, r) ∩ B = ∅ ∧ Bg(x, r) ∩ B = ∅Then A ⊂ B ⇒ a(A) ⊂ a(B)Q.E.D.Remark. This pretopology is not a VD one because, generallyspeaking, V(x) is not a filter of subsets of E.

As in the case of pretopologies using a given threshold for r,we can use the concepts of upper balls and lower balls to definetwo other pretopological structures on E. Let us considerB(x) = {Binf (x, r), r > 0} and B(x) = {Bsup(x, r), r > 0}.We can put:

a(A) = {x ∈ E/Binf (x, r) ∩ A = ∅,∀r, r > 0}and

a(A) = {x ∈ E/Bsup(x, r) ∩ A = ∅,∀r, r > 0}Obviously, we define two pseudo closures of two pretopologiesrespectively noted P and P . We obviously get the followingresult:

Property 9: If P denotes the pretopology induced by thequasipseudometric d, we have : P ≺ P ≺ PRemark. The two pretopologies P and P are VD ones.An interesting particular case is the case when d fulfills thetriangle axiom although being non symmetric. In that case,what happens to the pretopology induced by d ?

Property 10: If d fulfills the triangle axiom, the pseudoclosure function a of the pretopology P induced by d is anidempotent function.

Proof.We have to show that ∀A,A ⊂ E, a(a(A)) = a(A). In fact,it is sufficient to prove that ∀A,A ⊂ E, a(a(A)) ⊂ a(A).Let us consider x ∈ a(a(A)). Then

∀r, r > 0,∃yr, yr ∈ a(A), d(x, yr) < r∧∃y′r, y

′r ∈ a(A), d(y′

r, x) < r

But yr ∈ a(A) ∧ y′r ∈ a(A), then:

∀s, s > 0,∃zs, zs ∈ A, d(yr, zs) < s∧∃z′s, z′s ∈ A, d(z′s, yr) < s

By the triangle axiom, we can say:

∀r,∀s,∃zs, zs ∈ A, d(x, zs) < r+s∧∃z′s, z′s ∈ A, d(z′s, x) < r+s

It is sufficient to prove that x ∈ a(A)and then ∀A,A ⊂E, a(a(A)) ⊂ a(A).Q.E.D.Now, we have defined a family of pretopologies on E : thepretopologies associated to the quasipseudometric d for agiven value of r. We also have the pretopology induced onE by the quasipseudometric d. What are the links betweenall these pretopologies? The answer is given by the followingresult.

Property 11: ∀A,A ⊂ E, a(A) =⋂

r>0 ar(A)The pretopology P induced by the quasipseudometric d is thelower bound of the pretopologies Pr.

Proof.Let us consider x, x ∈ a(A)⇒ ∀r, r > 0, Bd(x, r) ∩ A = ∅ and Bg(x, r) ∩ A = ∅⇒ ∀r, r > 0, x ∈ ar(A)Q.E.D.

The following definition and result allows to characterizeelements of a(A).

Definition 9: Let (E, d) a quasipseudometric space, A asubset of E. For every x in E, we put:d(x,A) = inf{d(x, y), y ∈ A}d(A, x) = inf{d(y, x), y ∈ A}We can then write:

Property 12: Let A ∈ P(E), the two following assertionsare equivalents(i) x ∈ a(A)(ii) d(x,A) = 0 and d(A, x) = 0

Proof.x ∈ a(A)⇔ ∀r, r > 0, Bd(x, r) ∩ A = ∅ and Bg(x, r) ∩ A = ∅⇔ ∀r, r > 0,∃yr, yr ∈ A, 0 ≤ d(x, yr) < r and∀r, r > 0,∃y′

r, y′r ∈ A, 0 ≤ d(y′

r, x) < r⇔ inf{d(x, y), y ∈ A} = 0 and inf{d(y, x), y ∈ A} = 0⇔ d(x,A) = 0 and d(A, x) = 0Q.E.D.

A consequence of this result is that it is possible to charac-terize the neighborhoods of x by means of the quasipseudo-metric d:

Property 13: Let V ∈ P(E), a necessary and sufficientcondition for V to be a neighborhood of x is that d(x, V c) = 0or d(V c, x) = 0

Proof.It is sufficient to note that if V is a neighborhood of x, thatmeans that x does not belong to a(V c).Q.E.D.

IV. EQUIVALENT QUASIPSEUDOMETRICS

In this section, we shall examine the problem of quasipseu-dometrics which are ”equivalent” in relation with the pre-topologies induced by these quasipseudometrics.

Definition 10: Let d1 and d2 two quasipseudometrics on asame set E. If:∃α, α > 0,∃β, β > 0 such as ∀(x, y) ∈ E2, αd2(x, y) ≤d1(x, y) ≤ βd2(x, y), we shall say that d1 and d2 are (fromthe point of view of the topology) equivalent.

Let us now consider two quasipseudometrics d1 and d2 whichare equivalent. Let us denote by a1(A) the pseudoclosure ofany subset A of E by the pretopology induced by d1 andby a2(A) the pseudoclosure of any subset A of E by thepretopology induced by d2. According to the above definition,it is obvious that:

∀A,A ⊂ E, a1(A) = a2(A).So, we can say:

Property 14: If d1 and d2 are two equivalent quasipseudo-metrics on E, then the pretopologies induced on E by d1 andd2 are the same one.

V. SEPARABILITY

In this section, we shall give a necessary and sufficient con-dition for a quasipseudometric space (E, d) to be separated.This condition is the following:

Property 15: If the quasipseudometric d fullfills the triangleaxiom, then (E, d) is a separated space as a pretopologicalspace endowed with the pretopology induced on E by d.

Proof.Let us suppose that d fulfills the triangle axiom and thefollowing assertion is true:∃x, x ∈ E,∃y, y ∈ E, x = y, V (x) ∩ V (y) = ∅ whereV (x) ∈ V(x) and V (y) ∈ V(y).Then : ∀V (x),∀V (y),∃z, z ∈ V (x)∩ V (y). This will be truefor :V (x) = Bd(x, ε

2 ),∀ε

V (y) = Bg(x, ε2 ),∀ε

⇒ ∀ε > 0,∃zε, d(x, zε) < ε2 and d(zε, y) < ε

2⇒ ∀ε > 0, d(x, y) < ε by the triangle axiom.This leads us to x = y which is in contradiction with ourhypothesis.Q.E.D.

VI. CONCLUSION

The results presented in this paper are the first results ofa complete analysis on quasipseudometrics and on relatedpretopological spaces. Other theoretical questions are again tosolve, in particular the analysis of the conditions for that a Vpretopological space can be associated to a quasipseudometricspace. Otherwise, concerning the applications in the field ofdata analysis, it remains to define clustering methods foundedon that concept of quasipseudometric.

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