Topology and its Applications 160 (2013) 1794–1801
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Quasi-metric tree in T0 -quasi-metric spaces Olivier Olela Otafudu 1 School of Mathematical Sciences, North-West University (Mafikeng campus), Mmabatho 2735, South Africa
a r t i c l e
i n f o
Article history: Received 24 May 2013 Received in revised form 11 July 2013 Accepted 11 July 2013 MSC: 54E15 54E35 54C15 54E55 54E50
a b s t r a c t In his well-known paper dealing with metric trees and tight extension of metric spaces, Dress studied the relationship between metric trees and hyperconvex metric spaces. In the present work, we introduce similarly the notion of quasi-metric tree of a T0 -quasi-metric space. In particular, we show that large parts of the theory of metric trees do not use the symmetry of the metric and, under appropriate modifications, it still holds essentially unchanged for T0 -quasi-metrics. © 2013 Elsevier B.V. All rights reserved.
Keywords: Metric interval Metric tree T0 -quasi-metric Quasi-metric interval Quasi-metric tree
1. Introduction A metric space (X, m) is called a metric tree (see [2, p. 321]) if X satisfies (T 1) and (T 2). (T 1) is defined as for all x, y, z ∈ X, there exists a unique isometric embedding ϕ = ϕxy of the closed interval [0, m(x, y)] ⊆ R into X such that ϕ(0) = x and ϕ(m(x, y)) = y and (T 2) is defined as for any injective continuous map ϕ : [0, 1] → X : t → xt of the unit interval [0, 1] ⊆ R into X and any t ∈ [0, 1] one has m(x0 , xt ) + m(xt , x1 ) = m(x0 , x1 ). Let us also recall that a metric space is called hyperconvex (see e.g. [5, p. 78]) if and only if it is injective in the category of metric spaces and non-expansive maps. Dress [2] later rediscovered this concept and provided an independent approach invoking tight extensions. In analogy to Isbell’s theory, Kemajou et al. [6] proved that each T0 -quasi-metric space X has a q-hyperconvex hull QX , which is joincompact if X is joincompact. They called a T0 -quasi-metric space E-mail address:
[email protected]. The author would like to thank the Faculty Research Committee of the Faculty of Agriculture Science and Technology for partial financial support. 1
0166-8641/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.topol.2013.07.009
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q-hyperconvex if and only if it is injective in the category of T0 -quasi-metric spaces and non-expansive maps. In this paper, we generalize results from [2] on metric trees to T0 -quasi-metric spaces. The results of the present paper will be applied in future investigations by the author regarding quasi-metric trees in q-hyperconvex T0 -quasi-metric spaces [8]. Hence, it was necessary to develop the theory discussed below in detail although it sometimes closely follows the theory developed for classical metrics. 2. Preliminaries This section recalls the most important definitions that we shall use in the following. Definition 1. Let X be a set and let d : X × X → [0, ∞) be a function mapping into the set [0, ∞) of the nonnegative reals. Then, d is called a quasi-pseudometric on X if (a) d(x, x) = 0 whenever x ∈ X, (b) d(x, z) d(x, y) + d(y, z) whenever x, y, z ∈ X. We shall say that d is a T0 -quasi-metric provided that d also satisfies the following condition: For each x, y ∈ X, d(x, y) = 0 = d(y, x) implies that x = y. Remark 1. Let d be a quasi-pseudometric 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 a quasi-pseudometric, called the conjugate quasi-pseudometric of d. As usual, a quasi-pseudometric d on X such that d = d−1 is called a pseudometric. Note that for any (T0 -)quasi-pseudometric d, ds = max{d, d−1 } = d ∨ d−1 is a pseudometric (metric). Remark 2. We remark that for a quasi-pseudometric space (X, d): 1. For each x ∈ X and > 0, Bd (x, ) = {y ∈ X: d(x, y) < } denotes the open -ball at x. 2. The collection of all “open” balls yields a base for a topology τ (d). It is called the topology induced by d on X. 3. Similarly we set for each x ∈ X and 0, Cd (x, ) = {y ∈ X: d(x, y) }. Note that Cd (x, ) is τ (d−1 )-closed, but not τ (d)-closed in general. ˙ b = max{a − b, 0}. Note that u(x, y) = x − ˙ y with x, y ∈ R defines a T0 If a, b ∈ R, we shall put a − quasi-metric on the set R of the reals. A map f : (X, d) → (Y, e) between two quasi-pseudometric spaces (X, d) and (Y, e) is called an isometry provided that e(f (x), f (y)) = d(x, y) whenever x, y ∈ X. Two quasi-pseudometric spaces (X, d) and (Y, e) will be called isometric provided that there exists a bijective isometry f : (X, d) → (Y, e). A map f : (X, d) → (Y, e) between two quasi-pseudometric spaces (X, d) and (Y, e) is called non-expansive provided that e(f (x), f (y)) d(x, y) whenever x, y ∈ X. For further basic concepts used from the theory of asymmetric topology, we refer the reader to [3] and [7]. In addition, a recent preprint by Agyingi et al. [1] investigated the tight extensions of T0 -quasi-metric spaces. Herrmann and Moulton [4] found that tight-spans could be defined for more general maps such as directed metrics and distances. They showed that all these tight-spans as well as some related constructions can be defined in terms of point configurations. 3. Quasi-pseudometric interval The following definition is useful in this paper.
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Definition 2. Let (X, d) be a quasi-pseudometric space. For x, y ∈ X we define a subset x, y d of X containing x and y by x, y d = z ∈ X: d(x, z) + d(z, y) = d(x, y) .
Then x, y d is called a quasi-pseudometric interval of (X, d). Note that if (X, d) is a T0 -quasi-metric space which has only one point x, then x, x d contains at least the point x. Observe also that we obtain the standard definition of a metric interval in the case that (X, d) is a metric space (compare [2, p. 360]). Remark 3. Observe that for x, y ∈ X, we have that x, y d−1 if and only if y, x d . Example 1. (Compare [1, Example 1].) Consider the four point set X = {1, 2, 3, 4}. Let the T0 -quasi-metric q be defined by the distance matrix ⎛ ⎞ 0121 ⎜1 0 1 2⎟ ⎜ ⎟ M =⎜ ⎟ ⎝2 1 0 1⎠ 2110 that is, qi,j = q(i, j) whenever i, j ∈ X. One can check easily that q is a T0 -quasi-metric on X. Furthermore, it is readily checked that 2 ∈ 1, 3 q , 2 ∈ 3, 1 q and 2 ∈ 4, 1 q . Observe also that 2 ∈ 3, 4 q but q(1, 2) + q(2, 3) + q(3, 4) = q(1, 4), since 3 ∈ / 1, 4 q . Lemma 1. Let (X, d) be a quasi-pseudometric space. If z ∈ x, y d and y ∈ x, t d , then z ∈ x, t d and y ∈ z, t d . (Hence d(x, z) + d(z, y) + d(y, t) = d(x, t).) Proof. We have that d(x, t) d(x, z) + d(z, t) d(x, z) + d(z, y) + d(y, t) = d(x, y) + d(y, t) = d(x, t) imply d(x, t) = d(x, z) + d(z, t) and d(z, t) = d(z, y) + d(y, t).
2
Proposition 1. (Compare [1, Proposition 1].) Let (X, d) be a T0 -quasi-metric space. Fix y ∈ X. We define y in X by a1 y a2 if a1 ∈ y, a2 d . Then y is partial order on X. Proof. Indeed, y is reflexive since d(y, a) + d(a, a) = d(y, a). Let a1 , a2 ∈ X be such that a1 y a2 and a2 y a1 . Then d(y, a1 ) + d(a1 , a2 ) = d(y, a2 ) and d(y, a2 ) + d(a2 , a1 ) = d(y, a1 ). Then (y, a1 ) + d(a1 , a2 ) + d(a2 , a1 ) = d(y, a2 ) + d(a2 , a1 ) = d(y, a1 ). Furthermore d(a1 , a2 ) + d(a2 , a1 ) = 0. Therefore a1 = a2 by T0 -property of d. Hence y is antisymmetric. Suppose that a1 ∈ y, a2 d and a2 ∈ y, a3 d . It follows from Lemma 1 that a1 ∈ y, a3 d . Thus y is transitive. 2 Lemma 2. Let (X, d) be a T0 -quasi-metric space. For any x, y, z ∈ X if z ∈ x, y d then x, z d ∩z, y d = {z} and x, z d ∪ z, y d ⊆ x, y d . Proof. Let z ∈ x, y d then d(x, z) + d(z, y) = d(x, y). (a) Consider a ∈ x, z d ∩ z, y d . We have d(x, a) + d(a, z) = d(x, z)
(1)
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d(z, a) + d(a, y) = d(z, y).
(2)
and
By combining (1) and (2) we have d(x, a) + d(a, z) + d(z, a) + d(a, y) = d(x, z) + d(z, y). If x = a then d(x, a) + d(a, z) + d(z, a) + d(a, y) = d(x, z) + d(z, y) = d(x, y). If x = a, then d(a, a) + d(a, z) + d(z, a) + d(x, y) = d(x, y) that implies that d(a, z) + d(z, a) = 0,
(3)
but d(a, z) 0 and d(z, a) 0 we have that d(a, z) = 0 = d(z, a). By T0 property of (X, d) we have z = a. (b) Consider a ∈ x, z d ∪ z, y d . We show that a ∈ x, y d . The case a ∈ x, z d and a ∈ z, y d is obvious. If a ∈ x, z d we have d(x, a) + d(a, z) = d(x, z). Then by triangle inequality d(x, y) d(x, a) + d(a, z) + d(z, y) = d(x, z) − d(a, z) + d(a, z) + d(z, y). Furthermore d(x, y) d(x, a) + d(a, z) d(x, z) + d(z, y) = d(x, y). Therefore d(x, y) = d(x, a) + d(a, z). The case a ∈ z, y d is similar to the case a ∈ x, z d .
2
Proposition 2. (Compare [1, Proposition 3].) Let (X, d) be a joincompact T0 -quasi-metric space. Then for any y, a ∈ X with d(y, a) > 0 be given there exists e ∈ X such that a ∈ y, e d . Moreover there exists s ∈ X such that y ∈ s, a d . Proof. Consider Iy,a = {b ∈ X: a ∈ y, b d }. We have Iy,a = ∅, since a ∈ Iy,a . Then Iy,a equipped with the restriction of partial order y on X is a partially ordered set. Let K ⊆ Iy,a be a nonempty chain. We consider the net xk = k where k ∈ K, which is directed by the linear order of the chain K. Since (X, τ (ds )) is compact, we know that there is a subnet (xke )e∈E of (xk )k∈K converging to some point x in (X, ds ). We are going to show that x is an upper bound of K in Iy,a : Indeed for each e ∈ E we have that d(y, a) + d(a, xke ) = d(y, xke ) whenever e ∈ E, since xke ∈ Iy,a whenever e ∈ E. Thus a ∈ y, x d and x ∈ Iy,a . Since (xke )e∈E is a subnet of (xk )k∈K we see that ky ke eventually. By definition of y , if ky ke then we have d(x, xk ) + d(xk , xke ) = d(x, xke ). Taking limits in R where R is equipped with its usual topology, we have d(y, a) + d(a, x) = d(y, x), since |d(a, xke ) − d(a, xke )| ds (xke , x) and |d(y, xke ) − d(y, x)| ds (xke , x) whenever e ∈ E. Taking the limit as above, we get for each k ∈ K that d(y, xk ) + d(xk , x) = d(y, x). Consequently for each k ∈ K, xk ∈ y, x d . Thus x is an upper bound of K in Iy,a . Hence by Zorn’s lemma, Iy,a has a maximal element xmax . We have to show that a ∈ y, xmax d . Observe that d(y, xmax ) d(y, a) > 0. Suppose now that for some x ∈ X we have that xmax ∈ y, x d . Since a ∈ y, xmax d , we know that a y xmax y x. Thus a ∈ y, x d and therefore x ∈ Iy,a and xmax = x by minimality of xmax in Iy,a . Hence a ∈ y, xmax d . Applying the analogous argument to Jy,a = {b ∈ X: a ∈ y, b d−1 } we obtain the part of the statement dealing with y ∈ s, a d . 2
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Corollary 1. Let (X, d) be a joincompact T0 -quasi-metric space and let y, a ∈ X with d(y, a) > 0 be given. We equipped Iy,a = {b ∈ X: a ∈ y, b d } with the restriction of the partial order y (see Proposition 1). Then a ∈ y, xmax d . Corollary 2. Let (X, d) be a joincompact T0 -quasi-metric space. Moreover let two distinct points y1 , y2 ∈ X with d(y1 , y2 ) > 0 be given. Then there exist two points s, e ∈ X such that d(s, y1 ) + d(y1 , y2 ) + d(y2 , e) = d(s, e). Proof. By Proposition 2 there exists s ∈ X such that y1 ∈ s, y2 d . In particular d(s, y2 ) > 0. Again by Proposition 2 there exists e ∈ X such that y2 ∈ s, e d . By Lemma 1 we have d(s, y1 ) + d(y1 , y2 ) + d(y2 , e) = d(s, e). 2 4. Quasi-metric tree In this section we generalize some crucial results about metric tree space from [2] to our quasi-metric setting. Definition 3. Let (X, d) be a T0 -quasi-metric space. Then (X, d) is defined to be a (quasi-metric) tree or directed tree, if it satisfies the following two conditions: (QMT1) For any x, y ∈ X, there exists a unique function pair
ϕ = ϕxy = (ϕxy )1 , (ϕxy )2
where (ϕxy )1 :
0, d(x, y) , u−1 −→ (X, d)
is an isometry embedding such that (ϕxy )1 (0) = x and (ϕxy )1 (d(x, y)) = y, and (ϕxy )2 :
0, d−1 (x, y) , u −→ X, d−1
is an isometry embedding such that (ϕxy )2 (0) = y and (ϕxy )2 (d−1 (x, y)) = x. (QMT2) For any pair ϕ = (ϕ1 , ϕ2 ), where ϕi is injective continuous function from [0, 1] into X (i = 1, 2) such that ϕi : [0, 1] −→ X : t → xt one has d(x0 , xt ) + d(xt , x1 ) = d(x0 , x1 ). Remark 4. Note that in the above definition if d = d−1 the (X, d) is a metric space and our definition coincides with the definition of a metric tree in the sense of Dress (see [2, p. 321]). One can show the following result. Proposition 3. Let (X, d) be a T0 -quasi-metric space. Then (X, d) is quasi-metric tree if and only if (X, d−1 ) is quasi-metric tree. Proposition 4. Let (X, d) be a T0 -quasi-metric space. For any pair ϕ = (ϕ1 , ϕ2 ) of isometries such that ϕ1 : ([0, t], u−1 ) −→ (X, d) with, ϕ1 (0) = x and ϕ1 (t) = y and ϕ2 : ([0, t], u) −→ (X, d−1 ) with, ϕ2 (0) = y and ϕ2 (t) = x, we have that
ϕ1 [0, t] ⊆ x, y d
and
(fx )1 ◦ ϕ1 = (fx )1 |x,yd ◦ ϕ1 = Id [0,t]
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and
ϕ2 [0, t] ⊆ x, y d−1
and
(fx )2 ◦ ϕ2 = (fx )2 |x,yd−1 ◦ ϕ2 = Id [0,t] .
Proof. Let z ∈ ϕ1 ([0, t]) then there exists t1 ∈ [o, t] such that ϕ1 (t1 ) = z. If t1 = 0 then ϕ1 (0) = x ∈ x, y d . If t1 = t then ϕ1 (t) = y ∈ x, y d . If 0 < t1 < t, since ϕ1 is an isometry we have
d ϕ1 (0), ϕ1 (t1 ) + d ϕ1 (t1 ), ϕ1 (t) = u−1 (0, t1 ) + u−1 (t1 , t) = t = u−1 (0, t)
that implies that
d ϕ1 (0), ϕ1 (t1 ) + d ϕ1 (t1 ), ϕ1 (t) = d ϕ1 (0), ϕ1 (t) .
Therefore ϕ1 (t1 ) ∈ ϕ1 (0), ϕ1 (t) d = x, y d .
Hence ϕ1 ([0, t]) ⊆ x, y d . Similarly one shows that ϕ2 ([0, t]) ⊆ x, y d−1 for any t1 ∈ [0, t]. We observe that for any t1 ∈ [0, t] we have (fx )1 (ϕ1 (t1 )) = t1 and similarly for any t2 ∈ [0, t], we have (fx )2 (ϕ2 (t1 )) = t2 . 2 Proposition 5. Let (X, d) be a T0 -quasi-metric space. Then (X, d) satisfies condition (QMT1) in Definition 3 if and only if
(fx )1 |x,yd : x, y d , d → 0, d(x, y) , u−1
and
(fx )2 |x,yd− 1 : x, y d−1 , d−1 → 0, d(x, y) , u
are bijective isometries for all x, y ∈ X. Proof. Assume that (X, d) satisfies the condition (QMT1). Let us show that for z ∈ x, y d ⊆ X implies (ϕxy )1 (d(x, z)) = z. We have 0 d(x, z) d(x, y) since z ∈ x, y d ⊆ X. Then (fx )1 (z) = d(x, z) ∈ [0, d(x, y)] that implies that
(ϕxy )1 (fx )1 (z) = (ϕxy )1 d(x, z) = z
if z = x and z = y.
So (ϕxy )1 is the right inverse of (fx )1 |x,yd and by Proposition 4 is also the left inverse of (fx )1 |x,yd . But if z ∈ x, y d and if
ψ1 : 0, d(x, y) → x, z d ∪ z, y d ⊆ x, y d
is defined by ψ1 (t) = (ϕxz )1 (t) for 0 t d(x, z),
ψ1 (t) = (ϕzy )1 t − d(x, z) for d(x, z) t d(z, y),
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then ψ1 satisfies
ψ1 (0) = x, ψ1 d(x, z) = t, ψ1 d(x, y) = y
and
˙ t1 d ψ1 (t1 ), ψ1 (t2 ) = u−1 (t1 , t2 ) = t2 −
for t1 , t2 ∈ 0, d(x, z) or t1 , t2 ∈ d(x, z), d(x, y) ,
as well as for 0 t1 d(x, z) t2 d(x, y) we have
d ψ1 (t1 ), ψ1 (t2 ) d ψ1 (t1 ), ψ1 d(x, z) + d ψ1 d(x, z) , ψ1 (t2 )
= u−1 t1 , d(x, z) + u−1 d(x, z), t2 = d(x, z) − t1 + t2 − d(x, z) = u−1 (t1 , t2 ).
Then
˙ t1 . d ψ1 (t1 ), ψ1 (t2 ) u−1 (t1 , t2 ) = t2 −
(4)
Furthermore for 0 t1 d(x, z) t2 d(x, y) we have
d(x, y) d ψ1 (0), ψ1 (t1 ) + d ψ1 (t1 ), ψ1 (t2 ) + d ψ1 (t2 ), ψ1 d(x, y)
= t1 + d ψ1 (t1 ), ψ1 (t2 ) + d(x, y) − t2
that implies
t2 − t1 d ψ1 (t1 ), ψ1 (t2 ) .
Then
˙ t1 d ψ1 (t1 ), ψ1 (t2 ) . t2 −
(5)
Therefore from (4) and (5) we have
˙ t1 d ψ1 (t1 ), ψ1 (t2 ) = t2 −
for t1 , t2 ∈ 0, d(x, y) .
Thus ψ1 = (ϕxy )1 that in turn implies (ϕxy )1 (d(x, z)) = ψ1 (d(x, z)) = z. By similar arguments one shows that for z ∈ x, y d−1 ⊆ X implies (ϕxy )2 (d−1 (z, x)) = z. Vice versa, if
(fx )1 |x,yd : x, y d , d → 0, d(x, y) , u−1
is bijective isometry, then its inverse ϕ1 = (ϕx,y )1 : [0, d(x, y)] → x, y d ⊆ X is an isometry with ϕ1 (0) = x and ϕ1 (d(x, y)) = y and by Proposition 4, it is the only such isometry: if ϕ1 : [0, d(x, y)] → X is another isometry with ϕ1 (0) = x and ϕ1 (d(x, y)) = y, then (fx )1 |x,yd ◦ ϕ1 = (fx )1 |x,yd ◦ ϕ1 implies that ϕ1 = ϕ1 since (fx )1 |x,yd is injective. Similarly one verifies that ϕ2 is the only inverse of (fx )2 |x,yd−1 with ϕ2 = (ϕx,y )2 : [0, d−1 (x, y)] → x, y d−1 ⊆ X being an isometry with ϕ2 (0) = y and ϕ1 (d−1 (x, y)) = x. We have shown that (X, d) satisfies condition (QMT1) in Definition 3. 2
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Acknowledgement The author would like to thank the referee for several suggestions that have clearly improved the presentation of this paper. References [1] C.A. Agyingi, P. Haihambo, H.-P. Künzi, Endpoints in T0 -quasi-metric spaces, submitted for publication. [2] A.W.M. Dress, Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces, Adv. in Math. 53 (1984) 321–402. [3] P. Fletcher, W.F. Lindgren, Quasi-Uniform Spaces, Dekker, New York, 1982. [4] S. Herrmann, V. Moulton, Tress, tight-spans and point configuration, Discrete Math. 312 (16) (2012) 2506–2521. [5] M.A. Khamsi, W.A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory, John Wiley, New York, 2001. [6] E. Kemajou, H.-P.A. Künzi, O.O. Otafudu, The Isbell-hull of a di-space, Topology Appl. 159 (2012) 2463–2475. [7] H.-P.A. Künzi, Introduction to quasi-uniform spaces, in: F. Mynard, E. Pearl (Eds.), Beyond Topology, in: Contemp. Math., vol. 486, Amer. Math. Soc., 2009, pp. 239–304. [8] O.O. Otafudu, Quasi-metric trees and q-hyperconvex hull T0 -quasi-metric spaces, in preparation.