Yet Another Generalization of The Notion of a Metric Space
YYET ANOTHER GENERALIZATION OF THE NOTION OF A METRIC SPACE
SEYED MOHAMMAD AMIN KHATAMI
Department of Computer Science, Birjand University of Technology, Birjand, Iran
MADJID MIRZAVAZIRI
Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775,Iran
Abstract.
A generalization of the triangle inequality is introduced by a mapping similar to a t-conormmapping. This generalization leads us to a notion for which we use the (cid:63) -metric terminology. We areinterested in the topological space induced by a (cid:63) -metric. Considering some examples of non-trivial (cid:63) -metrizable topological spaces, we also study the product topology for a finite family of (cid:63) -metrizabletopological spaces.
Keywords:
Metric space, Generalization of metric space, Metric topology Introduction.
The familiar notion of a metric which seems to be introduced firstly by the French mathematicianFr´echet [2], is a mathematics model for distance. A metric, which is expected as a distance mapping ona nonempty set M , is defined by a mapping d : M → [0 , ∞ ) satisfying the following axioms:(M1) (identity of indiscernibles) ∀ x ∀ y ( d ( x, y ) = 0 ⇔ x = y ),(M2) (symmetry) ∀ x ∀ y ( d ( x, y ) = d ( y, x )),(M3) (triangle inequality) ∀ x ∀ y ∀ z ( d ( x, y ) ≤ d ( x, z ) + d ( z, y )).There are different generalizations of the notion of a metric. Pseudometric is a generalization in whichthe distance between two distinct points can be zero [6]. Metametric is a distance in which identicalpoints do not necessarily have zero distance [13]. Quasimetric is defined by omitting the symmetricproperty of the metric mapping [14]. Semimetric is defined by omitting the triangle inequality [15].Ultrametric is a metric with the strong triangle inequality ∀ x ∀ y ∀ z ( d ( x, y ) ≤ max { d ( x, z ) , d ( z, y ) } ) [11].Probabilistic metric is a fuzzy generalization of a metric where the distance instead of non-negative realnumbers is defined on distribution functions [7, 8]. There are many other extensions of the concept of ametric which have appeared in literatures (e.g. see [1, 4, 9, 3]).This paper is about a generalization of the notion of a metric, which is called a (cid:63) -metric, by spreadingout the triangle inequality. We use a symmetric associative nondecreasing continuous function (cid:63) :[0 , ∞ ) → [0 , ∞ ) with the boundary condition a (cid:63) a called t-definer to extend the triangle inequality.The function (cid:63) is indeed an extension of a well-known function, namely t-conorm, to the set of non-negative real numbers. Continuity of (cid:63) implies the existence of a dual operator for it, called residua,which simplify the calculations of (cid:63) -metric functions such as metric functions. E-mail addresses : [email protected], amin [email protected], [email protected], [email protected] . a r X i v : . [ m a t h . GN ] S e p YET ANOTHER GENERALIZATION OF THE NOTION OF A METRIC SPACE (cid:63) -metric. Recall that a t-conorm is a symmetric associative binary operator on the closed unit interval whichis nondecreasing on both arguments satisfying S ( x,
0) = 0 for all x ∈ [0 , Definition 2.1.
A triangular definer or a t-definer is a function (cid:63) : [0 , ∞ ) → [0 , ∞ ) satisfying thefollowing conditions:(T1) a (cid:63) b = b (cid:63) a ,(T2) a (cid:63) ( b (cid:63) c ) = ( a (cid:63) b ) (cid:63) c ,(T3) a ≤ b implies a (cid:63) c ≤ b (cid:63) c and c (cid:63) a ≤ c (cid:63) b ,(T4) a (cid:63) a ,(T5) (cid:63) is continuous in its first component with respect to the Euclidean topology.Obviously, because of the commutativity of a t-definer (T1), its continuity in the first componentimplies its continuity in the second component. Furthermore, [5, Proposition 1.19] shows that a t-definer is a non-decreasing function (T3), so the continuity in its first component is equivalent to itscontinuity. Definition 2.2.
Let (cid:63) be a t-definer and M be a nonempty set. A (cid:63) -metric on M is a function d : M → [0 , ∞ ) satisfies the first two axioms of metric, (M1) and (M2), together with the (cid:63) -triangleinequality as follows.(M3 (cid:63) ) ( (cid:63) -triangle inequality) ∀ x ∀ y ∀ z ( d ( x, y ) ≤ d ( x, z ) (cid:63) d ( z, y )).In this case ( M, d ) is called a (cid:63) -metric space. Additionally, if (M1) is changed in to the weak form(M1’) ∀ x ∀ y ( x = y → d ( x, y ) = 0),then (cid:63) is called a (cid:63) -pseudometric. Example 2.3.
The most important continuous t-conorms are (cid:32)Lukasiewicz , Maximum, and Productt-conorms which are described by S L ( a, b ) = min { a + b, } , S m ( a, b ) = max { a, b } , and S π ( a, b ) = a + b − a.b .But a t-conorm is defined on the closed unit interval while a t-definer is defined on non-negative realnumbers. The most important t-definers are: • (cid:32)Lukasiewicz t-definer: a (cid:63) L b = a + b , • Maximum t-definer : a (cid:63) m b = max { a, b } .Obviously, an (cid:63) L -metric is actually a metric and an (cid:63) m -metric is an ultrametric.The following example shows that there are (cid:63) -metrics which are not metric. Example 2.4.
Clearly a (cid:63) p b = ( √ a + √ b ) is a t-definer. The function d ( a, b ) = ( √ a − √ b ) forms an (cid:63) p -metric on [0 , ∞ ) which is not a metric. Indeed, d ( a, b ) = (cid:16) √ a − √ b (cid:17) = (cid:16) √ a − √ c + √ c − √ b (cid:17) ≤ (cid:16)(cid:113) ( √ a − √ c ) + (cid:113) ( √ c − √ b ) (cid:17) = (cid:16)(cid:112) d ( a, c ) + (cid:112) d ( c, b ) (cid:17) = d ( a, c ) (cid:63) p d ( c, b ) , while d (1 ,
25) = 16 (cid:2) d (1 ,
16) + d (16 ,
25) = 9 + 1.
ET ANOTHER GENERALIZATION OF THE NOTION OF A METRIC SPACE 3
Note that one of the reasons that in Example 2.4 the function d does not form a metric is that a (cid:63) p b (cid:2) a + b . The following definition describes a partial order between t-definers. Definition 2.5.
Assume that (cid:63) and (cid:63) are two t-definers. If the inequality a (cid:63) b ≤ a (cid:63) b holds for all a, b ≥
0, then (cid:63) is called weaker than (cid:63) (or (cid:63) is called stronger than (cid:63) ) and denoted by (cid:63) ≤ (cid:63) . Remark . The Maximum t-definer (cid:63) m is the weakest t-definer. Let (cid:63) be an arbitrary t-definer and a, b ≥
0. Since a ≥ a (cid:63) b ≥ b (cid:63) b . Similarly a (cid:63) b ≥ a . So, a (cid:63) b ≥ max { a, b } . It seems thatthe strongest t-definer can not be specified. Example 2.7.
Consider t-definers in Example 2.3 and 2.4. Furthermore let (cid:63) s be defined by a (cid:63) s b = √ a + b . Then we have max { a, b } ≤ a (cid:63) s b ≤ a + b ≤ a (cid:63) p b . Remark . Clearly, for any two t-definers (cid:63) and (cid:63) , if (cid:63) ≤ (cid:63) then any (cid:63) -metric space is a (cid:63) -metricspace. In particular, an ultrametric space is a (cid:63) -metric space for any t-definer (cid:63) .The definition of the residuum of a t-conorm is the key point that we use t-conorm for introducingthe concept of t-definer. The residuum of a t-definer plays a role such as the role of minus operator foraddition operator. Definition 2.9.
Let (cid:63) be a t-definer. The residuum of (cid:63) is defined by a . → b = inf { c : c (cid:63) a ≥ b } .Note that b ∈ { c : c (cid:63) a ≥ b } and therefore . → uniquely exists. Furthermore, for any a, b, c ∈ [0 , ∞ ),(1) c ≥ a . → b if and only if c (cid:63) a ≥ b, which is called the residuation property of (cid:63) and . → . Lemma 2.10.
Let (cid:63) be a t-definer and . → be its residuum. Then(1) a . → b = min { c : c (cid:63) a ≥ b } ,(2) . → a = a ,(3) a . → b = 0 if and only if a ≥ b ,(4) a (cid:63) ( a . → b ) = max { a, b } ,(5) a . → b ≥ ( a . → c ) . → ( c . → b ) .(6) a . → b ≤ ( a . → c ) (cid:63) ( c . → b ) .Proof. (1) Let A = { c : c (cid:63) a ≥ b } and α = inf A . So there exists a non-increasing sequence { c n } ⊆ A such that lim c n = α . Now, continuity of (cid:63) in its first component implies that α (cid:63) a = (lim c n ) (cid:63) a = lim ( c n (cid:63) a ) ≥ b .So α ∈ A that is α = min A . (2) . → a = min { c : c (cid:63) ≥ a } = min { c : c ≥ a } = a . (3) If a ≥ b then a (cid:63) a ≥ b . So 0 ∈ { c : c (cid:63) a ≥ b } . Therefore a . → b = 0. Conversely if a . → b = 0then min { c : c (cid:63) a ≥ b } = 0 that is a = 0 (cid:63) a ≥ b . (4) If a ≥ b , then by (3) a . → b = 0 and therefore a (cid:63) ( a . → b ) = a (cid:63) a . If a ≤ b , then since (cid:63) is acontinuous function, a (cid:63) ( a . → b ) = a (cid:63) min { c : c (cid:63) a ≥ b } = min { a (cid:63) c : a (cid:63) c ≥ b } .Now, taking the continuous function f ( c ) = a (cid:63) c we have f (0) = a ≤ b = b (cid:63) ≤ b (cid:63) a = f ( b ), YET ANOTHER GENERALIZATION OF THE NOTION OF A METRIC SPACE therefore by the intermediate value theorem there exists some c ∈ [0 , b ] for which f ( c ) = b . So, min { a(cid:63)c : b ≤ a (cid:63) c } = b that is a (cid:63) ( a . → b ) = b . (5) Since (cid:63) is commutative and associative, by (4) we get( a . → b ) (cid:63) ( b . → c ) (cid:63) a = a (cid:63) ( a . → b ) (cid:63) ( b . → c ) ≥ b (cid:63) ( b . → c ) ≥ c Now, using the residuation property 1 two times on ( a . → b ) (cid:63) ( b . → c ) (cid:63) a ≥ c we get (5) . (6) By (4) a . → c ≥ ( c . → b ) . → ( a . → b ). So, residuation of (cid:63) and . → fulfills the proof. (cid:3) Example 2.11.
Let’s consider t-definers in Example 2.7. If a ≥ b then a . → b = 0 and if a < b thenfor (cid:32)Lukasiewicz t-definer ”+”: a . → b = b − a ,for Maximum t-definer: a . → b = b ,for (cid:63) s : a . → b = √ b − a ,for (cid:63) p : a . → b = ( √ b − √ a ) . Remark . Let (cid:63) be a t-definer and . → be its residuum. Define d : [0 , ∞ ) → [0 , ∞ ) by d ( a, b ) = ( a . → b ) (cid:63) ( b . → a ). Obviously, d satisfies (M2). Furthermore, Lemma 2.10 (
3) implies that d satisfies (M1)and Lemma 2.10 (
6) implies that d satisfies (M3 (cid:63) ), d ( a, b ) = ( a . → b ) (cid:63) ( b . → a ) ≤ ( a . → c ) (cid:63) ( c . → b ) (cid:63) ( b . → c ) (cid:63) ( c . → a )= ( a . → c ) (cid:63) ( c . → a ) (cid:63) ( b . → c ) (cid:63) ( c . → b )= d ( a, c ) (cid:63) d ( c, b ) . So, d forms a (cid:63) -metric on [0 , ∞ ). The induced (cid:63) -metrics of t-definers in Example 2.7 are as follows d L ( a, b ) = | b − a | forms an (cid:63) L -metric on [0 , ∞ ), d max ( a, b ) = (cid:26) a = b max { a, b } a (cid:54) = b forms an (cid:63) m -metric on [0 , ∞ ), d s ( a, b ) = (cid:112) | b − a | forms an (cid:63) s -metric on [0 , ∞ ), d p ( a, b ) = |√ b − √ a | forms an (cid:63) p -metric on [0 , ∞ ).Note that d L also defines a (cid:63) L -metric (or a metric) on R . Similarly, d s also defines a (cid:63) s -metric on R .3. Topology of (cid:63) -metric.
In this section, we extend some topological concepts of metric spaces to (cid:63) -metric spaces.
Definition 3.1.
Assume that (
M, d ) is a (cid:63) -metric space. For any a ∈ M and r >
0, the “open ballaround a of radius r ” is the set N r ( a ) = { b : d ( a, b ) < r } .For a subset A of M , a point x ∈ A is called an “interior point” of A if there exists (cid:15) > N (cid:15) ( x ) ⊂ A . A is said to be an “open” subset of M whenever any point of A is an interior point.The following theorem shows that the set of all open subsets of a (cid:63) -metric space ( M, d ) forms atopology on M called the (cid:63) -metric topology. Theorem 3.2.
For every (cid:63) -metric space ( M, d ) , the set of all open subsets of M forms a Hausdorfftopology on M , denoted by τ d . ET ANOTHER GENERALIZATION OF THE NOTION OF A METRIC SPACE 5
Proof.
Let τ d = { A ⊆ M : A is an open subset of M } . Obviously ∅ , M ∈ τ d . Assume that A, B ∈ τ d .Since ∅ ∈ τ d , if A ∩ B = ∅ there is nothing to prove. So, assume that A ∩ B (cid:54) = ∅ . We indicate that anypoint a ∈ A ∩ B is an interior point of A ∩ B . Since A and B are open sets, a is an interior point of A and B . So there exists r > s > N r ( a ) ⊆ A and N s ( B ) ⊆ B . If we set t = min { r, s } ,then N t ( a ) ⊆ N r ( a ) ∩ N s ( a ) ⊆ A ∩ B and so a is an interior point of A ∩ B . On the other hand, an easyargument shows that the union of arbitrary family of open sets is open.Now consider two distinct points a, b ∈ M . There exists s > such that s (cid:63) s < d ( a, b ). Indeed,otherwise d ( a, b ) < s (cid:63) s for any s > (cid:63) implies that d ( a, b ) = 0 which is acontradiction. Now, we show N s ( a ) ∩ N s ( b ) = ∅ which completes the proof. To this end, if there existssome c ∈ N s ( a ) ∩ N s ( b ) then we get the following contradiction: d ( a, b ) ≤ d ( a, c ) (cid:63) d ( c, b ) < s (cid:63) s < d ( a, b ). (cid:3) The notions and concepts of topological spaces such as “closed set”, “interior and closure of a set”,“limit point and the set of limit points of a set”, “continuous function”, and so forth are defined as usual(e.g. see [10] or [6]).The following theorem shows that in (cid:63) -metric spaces, open balls are open sets .
Lemma 3.3.
In every (cid:63) -metric space ( M, d ) , open balls are open sets.Proof. Assume that (cid:63) is a t-definer, . → is the residuum of (cid:63) , ( X, d ) is a (cid:63) -metric space, x ∈ M , and r >
0. We show that every y ∈ N r ( x ) is an interior point of N r ( x ). To this end for (cid:15) = d ( x, y ) . → r , weshow that N (cid:15) ( y ) ⊆ N r ( x ). For this consider z ∈ N (cid:15) ( y ). So , d ( z, y ) < (cid:15) that is d ( z, y ) < d ( x, y ) . → r .Now the residuation of . → and (cid:63) implies that d ( z, y ) (cid:63) d ( x, y ) < r and so by the (cid:63) -triangle inequality andsymmetric property of (cid:63) and d we have d ( x, z ) ≤ d ( x, y ) (cid:63) d ( y, z ) < r which show that z ∈ N r ( x ). (cid:3) Now, by Definition 3.1 and Lemma 3.3, for a (cid:63) -metric space (
M, d ) the set, B d = { N r ( a ) : a ∈ M and r > } is a base for the induced topology of d on M which is called the open ball base of τ d . Theorem 3.4.
Every (cid:63) -metric space ( M, d ) is first countable.Proof. Let a be an arbitrary point of M . We must show that there exists a countable family { U n } n ∈ N ofneighbourhoods of a such that every neighbourhood of a contain at least one of U n s. To this end for any n ∈ N set U n = N /n ( a ). By Lemma 3.3 any U n is a neighbourhood of a and the proof is complete. (cid:3) Theorem 3.5.
Every (cid:63) -metric space ( M, d ) is Normal.Proof. The proof is similar to the one for metric spaces (e.g. see [10, Theorem 32.2]). Let A and B betwo closed subset of M . Since B is a closed subset of M , for any a ∈ A let N r a ( a ) be an open ball suchthat N r a ( a ) ∩ B = ∅ . Similarly, for any b ∈ B the closeness of A implies that one could find N r b ( b ) suchthat N r b ( b ) ∩ A = ∅ . Now, for any a ∈ A and b ∈ B assume that s a and s b are such that s a (cid:63) s a < r a and s b (cid:63) s b < r b , respectively. Set, U = (cid:91) a ∈ A N s a ( a ) and V = (cid:91) b ∈ B N s b ( b ). U and V are open sets containing A and B respectively. Furthermore, we claim that U ∩ V = ∅ .Indeed, if c ∈ U ∩ V then there exists a ∈ A and b ∈ B such that c ∈ N s a ( a ) ∩ N s b ( b ) and therefore d ( a, b ) ≤ d ( a, c ) (cid:63) d ( c, b ) < s a (cid:63) s b . Now, without loss of generality we could assume that s b ≤ s a . So, d ( a, b ) < s a (cid:63) s a < r a which means that b ∈ N r a ( a ), a contradiction. (cid:3) YET ANOTHER GENERALIZATION OF THE NOTION OF A METRIC SPACE Product topology for (cid:63) -metric.
Recall that for a family { ( X i , τ i ) } i ∈ I of topological spaces, the product topology is the weakesttopology on X = (cid:81) i ∈ I X i which makes all of the projection maps { π i : X → X i } ı ∈ I continuous. Keepin mind that { (cid:81) i ∈ I U i : U i is open in X i and U i (cid:54) = X i for only finitely many i } is a base for theproduct topology on X . Furthermore, if for each i ∈ I the topology on X i is given by a basis B i then { (cid:81) i ∈ I B i : B i ∈ B i and B i (cid:54) = X i for only finitely many i } form a basis for product topology on X . Remark . If { ( M i , d i ) } ni =1 be a finite family of metric spaces, then the product topology on M = (cid:81) ni =1 M i is the same as the induced topology of the following three significant metrics on M = (cid:81) ni =1 M i (e.g. see [12, Theorem 4.5.1]). • (Maximum metric) d max (¯ x, ¯ y ) = max ≤ i ≤ n d i ( x i , y i ), • (Euclidean product metric) d E (¯ x, ¯ y ) = (cid:112)(cid:80) ni =1 d i ( x i , y i ) , • (Taxicab metric) d T (¯ x, ¯ y ) = (cid:80) ni =1 d i ( x i , y i ),The coming figure describes these three metrics and their corresponding open balls on R more precisely. Figure 1. d max , d E , and d T on R Definition 4.2.
For a (cid:63) -metric d on M , the (cid:63) -product topology on M n is the product topology inducedby the (cid:63) -metric topology of M .The following theorem demonstrates a situation similar to that of Remark 4.1 for (cid:63) -metric spaces. Theorem 4.3.
Let { ( M i , d i ) } ni =1 be a family of (cid:63) -metric spaces. Assume that M = (cid:81) ≤ i ≤ n M i anddefine d max and d T by d max (¯ x, ¯ y ) = max ≤ i ≤ n d i ( x i , y i ) , d T (¯ x, ¯ y ) = d ( x , y ) (cid:63) d ( x , y ) (cid:63) ... (cid:63) d n ( x n , y n ) .Then d max and d T define (cid:63) -metrics on M . Furthermore the induced topology of these two metrics on M is the same as the product topology on M .Proof. Obviously d max and d T satisfies the first two properties of (cid:63) -metric, namely ”identity of indis-cernibles” and ”symmetry”. ET ANOTHER GENERALIZATION OF THE NOTION OF A METRIC SPACE 7
For the (cid:63) -triangle inequality let ¯ x, ¯ y, ¯ z ∈ M . If max ≤ i ≤ n d i ( x i , y i ) = d k ( x k , y k ) for some 1 ≤ k ≤ n ,then we have d max (¯ x, ¯ y ) = max ≤ i ≤ n d i ( x i , y i )= d k ( x k , y k ) ≤ d k ( x k , z k ) (cid:63) d k ( z k , y k ) ≤ d max (¯ x, ¯ z ) (cid:63) d max (¯ z, ¯ y ) . The following argument shows that d T also admits the (cid:63) -triangle inequality. d T (¯ x, ¯ y ) = d ( x , y ) (cid:63) d ( x , y ) (cid:63) ... (cid:63) d n ( x n , y n ) ≤ (cid:0) d ( x , z ) (cid:63) d ( z , y ) (cid:1) (cid:63) (cid:0) d ( x , z ) (cid:63) d ( z , y ) (cid:1) (cid:63) ... (cid:63) (cid:0) d n ( x n , z n ) (cid:63) d n ( z n , y n ) (cid:1) = (cid:0) d ( x , z ) (cid:63) d ( x , z ) (cid:63) ... (cid:63) d n ( x n , z n ) (cid:1) (cid:63) (cid:0) d ( z , y ) (cid:63) d ( z , y ) (cid:63) ... (cid:63) d n ( z n , y n ) (cid:1) = d T (¯ x, ¯ z ) (cid:63) d T (¯ z, ¯ y ) . For the latter argument, firstly note that the induced topology of d max on M is as the same as theinduced topology of d T on M . Indeed, if we denote the elements of the open ball base of inducedtopologies of d T and d max by N Tr (¯ a ) and N maxr (¯ a ) respectively, then N Tr (¯ a ) ⊆ N maxr (¯ a ) ⊆ N T r (cid:63) r (cid:63) ... (cid:63) r (cid:124) (cid:123)(cid:122) (cid:125) n-times (¯ a ).Now, let B be a basis for the product topology and B = (cid:81) ni =1 N r i ( a i ) be an element of B and ¯ x ∈ B .Since for each 1 ≤ i ≤ n , x ∈ N r i ( a i ) and N r i ( a i ) is a (cid:63) -open set, there exists (cid:15) i > N (cid:15) i ( x i ) ⊆ N r i ( a i ). Let (cid:15) = min ≤ i ≤ n { (cid:15) i } . Obviously, N max(cid:15) (¯ x ) ⊆ B .On the other hand, if ¯ x ∈ N maxr (¯ x ) then for each 1 ≤ i ≤ n , x i ∈ N r ( a i ) and so there exists (cid:15) i > N (cid:15) i ( x i ) ⊆ N r ( a i ). Assuming (cid:15) = min ≤ i ≤ n { (cid:15) i } we get N (cid:15) (¯ x ) ⊆ N maxr (¯ a ). So the inducedtopology of d max is as the same as the product topology on M . (cid:3) References [1] Jack G. Ceder, Some generalizations of metric spaces,
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