Cartesian products of the g -topologies are a g -topology
Jumaev Davron Ilxomovich, Ishniyazov Baxrom Normamatovich, Tagaymuratov Abror Olimovich
aa r X i v : . [ m a t h . GN ] J un Cartesian products of the g -topologies are a g -topology Jumaev Davron Ilxomovich , Ishniyazov Baxrom Normamatovich , Tagaymuratov Abror Olimovich Abstract
We show that unlike the usual topologies the g -topologies are closed with re-spect to the Cartesian products. Moreover, we bring much detailed explanationssome examples of concepts related the statistical metric spaces.2010 Mathematics Subject Classification.
Key words and phrases:
Statistical metric space, g -topology, Cartesian product. As mentioned in [1] the three principal applications of statistical metrics are to macro-scopic, microscopic and physiological spatial measurements. Statistical metrics are de-signed to provide us firstly with a method removing conceptual difficulties from micro-scopic physics and transferring them into the underlying geometry, secondly with a treat-ment of thresholds of spatial sensation eliminating the intrinsic paradoxes of the classicaltheory.The notion of distance is defined in terms of functions, points and sets. Indeed, inmany situations, it is appropriate to look upon the distance concept as a statistical ratherthan a determinate one. More precisely, instead of associating a number to the distance d ( p, q ) with every pair of points p , q , one should associate a distribution function F pq andfor any positive number x , interpret F pq ( x ) as the probability that the distance from p toq be less than x .Using this idea, K. Menger in [1] defined a statistical metric space using the proba-bility function in the year 1942. In 1943, shortly after the appearance of Mengers article,Wald published an article [2] in which he criticized Mengers generalized triangle inequality.In [3] the following questions raised by Thorp in statistical metric spaces: • What are the necessary and sufficient conditions that the g -topology of type V tobe of type V D ? • What are the necessary and sufficient conditions that the g -topology of type V α tobe the g -topology of type V D ? • What conditions are both necessary and sufficient for the g -topology of type V α tobe a topology? Tashkent Institute of Architecture and Civil Engineering, 7, Kichik Xalqa yuli Str., Tashkent 100084, Uzbekistan,e-mail: [email protected] Tashkent State Agrarian University, 2, University Str., Kibray District, Tashkent Region 100700, Uzbekistan,e-mail: [email protected] Chirchik State Pedagogical Institute, 104, Amir Temur Str., Chirchik town, Tashkent Region 111700, Uzbekistan,e-mail: [email protected]
1n [4] it had given partial answer to the above questions. Also, it was provided thebasis for carrying out analysis in statistical metric spaces, in particular for the developmentof various g -topologies, neighborhoods defined in a statistical metric space and also theimprovement of λ Ω -open sets in a generalized metric space. The authors of [4] had givenmore examples of the neighborhoods defined in a statistical metric space and the specialkind of relationship between various g -topologies defined by Thorp in a SM space. Weseem that the examples have shortcomings. That is why we in the present paper completedthe shortcomings and give more detail clarifies.Further, we give positive answers to the question, expressed in [5], which asks: Is theCartesian product of the g -topological spaces a g -topological space? A statistical metric space ( SM space) is an ordered pair ( S, F ) where S is a non-null setand F is a mapping from S × S into the set of distribution functions (that is, real-valuedfunctions of a real variable which are everywhere defined, non decreasing, left-continuousand have infimum 0 and supremum 1).The distribution function F ( p, q ) associated with a pair of points p and q in S isdenoted by F pq . Moreover, F pq ( x ) represents the probability that the “distance” between p and q is less than x .The functions F pq are assumed to satisfy the following:( SM - I ) F pq ( x ) = 1 for all x > p = q .( SM - II ) F pq (0) = 0.( SM - III ) F pq = F qp .( SM - IV ) If F pq ( x ) = 1 and F qr ( y ) = 1, then F pr ( x + y ) = 1.We often find it convenient to work with the tails of the distribution functions ratherthan with these distribution functions themselves. Then the tail of F pq , denoted by G pq ,is defined by G pq ( x ) = 1 − F pq ( x ) for all x ∈ R . Let (
S, F ) be a statistical metric space. Then the Menger inequality is,( SM - IV m ) F pr ( x + y ) ≥ T ( F pq ( x ) , F qr ( y ))holds for all points p , q , r ∈ S and for all numbers x , y ≥ T is a 2-place functionon the unit square satisfying:( T - I ) 0 ≤ T ( a, b ) ≤ a, b ∈ [0 , T - II ) T ( c, d ) ≥ T ( a, b ) for all a, b, c, d ∈ [0 ,
1] such that c ≥ a , d ≥ b (monotonicity).( T - III ) T ( a, b ) = T ( b, a ) for all a, b ∈ [0 ,
1] (commutativity).( T - IV ) T (1 ,
1) = 1.( T - V ) T ( a, > a >
0. 2et (
S, F ) be a statistical metric space, p ∈ S and u , v be positive numbers. Then N p ( u, v ) = { q ∈ S : F pq ( u ) > − v } = { q ∈ S : G pq ( u ) < v } is called the ( u, v )-sphere with the center p .The following example shows the existence of ( u, v )-sphere in a statistical metricspace. Example 1
Consider the SM space ( S, F ) where S denotes the possible outcomes ofgetting a tail when a coin is tossed once. Then S = { , } . Let F pq ( u ) be the probabilitythat the “distance” between p and q is less than u where u > p, q ∈ S . We have(A. A. Zaitov): F ( u ) = F ( u ) = 1 for all u >
0; and F ( u ) = F ( u ) = ( , if 0 < u ≤ , , if u > . Fix p = 0. Then N ( u, v ) = { } , if 0 < u ≤ , ≤ v ≤ , { , } , if 0 < u ≤ , v > , { , } , if u > , v ≥ . Fix p = 1. Then N ( u, v ) = { } , if 0 < u ≤ , ≤ v ≤ , { , } , if 0 < u ≤ , v > , { , } , if u > , v ≥ . For fixed positive numbers u and v , define a set U ( u, v ) = { ( p, q ) ∈ S × S : G pq ( u ) < v } . Let us illustrate such sets in the following example.
Example 2
Consider the SM space ( S, F ) where S denotes the possible outcomes ofrolling a dice. Then S = { , , , , , } and the distribution function F pq ( x ) is theprobability that the “distance” between p and q is less than u where u > p, q ∈ S .Consider the usual metric on S induced on the real line, i. e. d ( p, q ) = | q − p | . We have(A. A. Zaitov): 3 pp ( u ) = 1 for all u > p = 1 , , , , , F p ( p +1) ( u ) = F ( p +1) p ( u ) = ( , < u ≤ , , u > , where p = 1 , , , , F p ( p +2) ( u ) = F ( p +2) p ( u ) = ( , < u ≤ , , u > , where p = 1 , , , F p ( p +3) ( u ) = F ( p +3) p ( u ) = ( , < u ≤ , , u > , where p = 1 , , F p ( p +4) ( u ) = F ( p +4) p ( u ) = ( , < u ≤ , , u > , where p = 1 , F p ( p +5) ( u ) = F ( p +5) p ( u ) = ( , < u ≤ , , u > , where p = 1 . Consequently:for every pair of u , v such that 0 < u ≤
1, 0 < v ≤ U ( u, v ) = { ( p, q ) ∈ S × S : d ( p, q ) = 0 } = { ( p, p ) : p ∈ S } = ∆( S );for every pair of u , v such that 1 < u ≤
2, 0 < v ≤ U ( u, v ) = { ( p, q ) ∈ S × S : d ( p, q ) ≤ } == ∆( S ) ∪ { ( p, p + 1) : p = 1 , , , , } ∪ { ( p + 1 , p ) : p = 1 , , , , } def = ∆ ( S );for every pair of u , v such that 2 < u ≤
3, 0 < v ≤ U ( u, v ) = { ( p, q ) ∈ S × S : d ( p, q ) ≤ } == ∆ ( S ) ∪ { ( p, p + 2) : p = 1 , , , } ∪ { ( p + 2 , p ) : p = 1 , , , } def = ∆ ( S );for every pair of u , v such that 3 < u ≤
4, 0 < v ≤ U ( u, v ) = { ( p, q ) ∈ S × S : d ( p, q ) ≤ } == ∆ ( S ) ∪ { ( p, p + 3) : p = 1 , , } ∪ { ( p + 3 , p ) : p = 1 , , } def = ∆ ( S );for every pair of u , v such that 4 < u ≤ , < v ≤ U ( u, v ) = { ( p, q ) ∈ S × S : d ( p, q ) ≤ } == ∆ ( S ) ∪ { ( p, p + 4) : p = 1 , } ∪ { ( p + 4 , p ) : p = 1 , } def = ∆ ( S );for every pair of u , v such that u >
5, 0 < v ≤ U ( u, v ) = { ( p, q ) ∈ S × S : d ( p, q ) ≤ } == ∆ ( S ) ∪ { ( p, p + 5) : p = 1 } ∪ { ( p + 5 , p ) : p = 1 } = S × S. Also, it is easy to see that U ( u, v ) = S × S for every u , v such that u > v > Z of ordered pairs of positive numbers, i. e. Z ⊂ (0 , + ∞ ) , let N ( Z ) = { N p ( u, v ) : ( u, v ) ∈ Z, p ∈ S } and U ( Z ) = { U ( u, v ) : ( u, v ) ∈ Z } . A non-null collection { N p } of subsets N p of a set S associated with a point p ∈ S isa family of neighborhoods for p if each N p contains p . Let the family of neighborhoods beassociated with each point p of a set S . V In this case the set S and the collection of neighborhoods is called the g -topologicalspace of type V [3].The closure of a subset E of S , written E , is the set of points p such that eachneighborhood of p intersects E . The interior of E is the complement of the closure of thecomplement of E . A g -topological space S is symmetric if, for every pair of points p and q , p is in { q } iff q is in { p } .E. Thorp introduced the following g -topologies in a statistical metric space ( S, F ). N is type V . N . For each point p and each neighborhood U p of p , there is a neighborhood W p of p such that for each point q of W p , there is a neighborhood U q of q contained in U p . N . For each point p and each pair of neighborhoods U p and W p of p , there is aneighborhood of p contained in the intersection of U p and W p .The following are various g -topologies in a statistical metric space ( S, F ) defined byE. Thorp. V D . If the conditions N and N are satisfied, then the collection of neighborhoods on S is called the g -topology of type V D . V α . The collection of neighborhoods on S is called the g -topology of type V α if theconditions N and N are satisfied. T op . A g -topology is a topology if the conditions N , N and N are satisfied.Let S be a set and ( P, < ) be a partially ordered set with least element 0. A generalized´ecart ( g -´ecart for short) is a mapping G : S × S → P. If a g -´ecart G satisfies G ( p, p ) = 0 and the set S consists of more than one point, the g -´ecart g -topology for S is the g -topology determined from G , and its partially orderedrange set P , as follows:For each f > P and each p ∈ S , the f -sphere for p is a set of the form N p ( f ) = { q ∈ S : G ( p, q ) < f } . Then for each p ∈ S , the collection of f -spheres N p ( P ) = { N p ( f ) : f > , p ∈ P } is a family of neighborhoods for p .The g -´ecart associated with a statistical metric space ( S, F ) is the mapping G definedby G ( p, q ) = G pq . 5 xample 3 Let S = N and P = N ∪ { } be a partially ordered set with the relation < where N denote the set of all positive integers. Let A = { , , } be a subset of S . Define G ( p, q ) = , if p / ∈ A, q ∈ S, , if ∈ A, q / ∈ S, , if / ∈ A, q / ∈ S. and for p ∈ A , q ∈ A define G ( p, q ) as follows: G (1 ,
1) = 0 , G (1 ,
2) = 2 , G (1 ,
3) = 3 ,G (2 ,
1) = 4 , G (2 ,
2) = 0 , G (2 ,
3) = 6 ,G (3 ,
1) = 1 , G (3 ,
2) = 2 , G (3 ,
3) = 0 .f -sphere for each p ∈ S has the following form (A. A. Zaitov): Case p = 1: N ( f ) = ∅ , if f = 0 , { } , if f = 1 ,S \ { , } , if f = 2 ,S \ { } , if f = 3 ,S, if f ≥ . Case p = 2: N ( f ) = ∅ , if f = 0 , { } , if f = 1 ,S \ { , } , if 2 ≤ f ≤ ,S \ { } , if 5 ≤ f ≤ ,S, if f ≥ . Case p = 3: N ( f ) = ∅ , if f = 0 , { } , if f = 1 ,S \ { } , if f = 2 ,S, if f ≥ . Case of arbitrary p ∈ S \ A : N p ( f ) = ∅ , if f = 0 ,S \ A, if f = 1 ,S, if f ≥ . Given a statistical metric space (
S, F ), for each pair of points p and r in S , number u >
0, the r -sphere with center p , N p ( r ; u ) is defined to be the sphere N p ( r ; u ) = N ( G pr ( u )) = { q : G pq ( u ) < G pr ( u ) } . R - g -topology for ( S, F ) is the structure whose family of neighborhoods at each point p is the collection N p ( r ) = { N p ( r ; u ) : r ∈ S, u > } . Example 4
Consider the SM space ( S, F ) where S = N and the distribution function F pq ( x ) = ( xd ( p, q ) , if 0 < x < d ( p, q ) , d ( p, q ) = 0 , , if x ≥ d ( p, q ) . where d ( p, q ) = | q − p | , p, q ∈ S .Fix p = 1 and r = 2 from S . Let x = . Then G pr ( x ) = G ( ) = 1 − F ( ) = 0 . F ( ) = 1 > .
25 and F q ( ) = ( q − ≤ .
25 for every q ≥ N (cid:18)
2; 14 (cid:19) = N (cid:18) G (cid:18) (cid:19)(cid:19) == (cid:26) q ∈ S : G q (cid:18) (cid:19) < . (cid:27) = (cid:26) q ∈ S : F q ( 14 ) > . (cid:27) = { } . Note that N ( G pr ) = ∅ if p = r . Really, for every u > p ∈ S one has F pp ( u ) = 1,consequently, G pp ( u ) = 0. Since 0 ≤ G pq ( u ) ≤ u > p, q ∈ S , there exists no q ∈ S such that G pq ( u ) < G pp ( u ). Hence N ( G pp ) = ∅ , p ∈ S . Remark 1
In a SM space ( S, F ), we use the following notations:( a ) Let τ denote the g -topology of type V .( b ) Let τ D denote the g -topology of type V D .( c ) Let τ α denote the g -topology of type V α .( d ) Let τ e denote the g -´ecart g -topology.( e ) Let τ R denote the R - g -topology.( f ) Each element in N ( X ) is called a τ -neighborhood.( g ) Each element in N p ( P ) is called a τ e -neighborhood.( h ) Each element in N p ( r ) is called a τ R -neighborhood. Let ( S , F ) and ( S , F ) be statistical metric spaces, ( p , p ), ( q , q ), ( r , r ) ∈ S × S .Put F p , p )( q , q ) ( x ) = F p q ( x ) · F p q ( x ) . (1)Obviously, conditions ( SM - I ) – ( SM - III ) are true. Let us show ( SM - IV ) is also satisfied.Assume that F p , p )( q , q ) ( x ) = 1 and F q , q )( r , r ) ( y ) = 1. These equalities mean that the7distance” between ( p , p ) and ( q , q ) less then x , and the “distance” between ( q , q )and ( r , r ) less than y . Then clearly, that the “distance” between ( p , p ) and ( r , r )less than x + y . Consequently, the probability that the “distance” between ( p , p ) and( r , r ) is less than x + y , is 1, i. e. F p , p )( r , q ) ( x + y ) = 1.So, (1) is defined correctly. Its tail G p , q )( p , q ) ( x ) = 1 − F p , q )( p , q ) ( x ) for all x ∈ R . Now we check the Menger inequality. Let T ( a, b ) = ab , a, b ∈ [0 , T - I ) – ( T - V ). Suppose F p r ( x + y ) ≥ T ( F p q ( x ) , F q r ( y )) and F p r ( x + y ) ≥ T ( F p q ( x ) , F q r ( y )) . Then F p , p )( r , r ) ( x + y ) = F p r ( x + y ) · F p r ( x + y ) ≥≥ T ( F p q ( x ) , F q r ( y )) · T ( F p q ( x ) , F q r ( y )) == F p q ( x ) · F q r ( y ) · F p q ( x ) · F q r ( y ) == F p , p )( q , q ) ( x ) · F q , q )( r , r ) ( y ) == T ( F p , p )( q , q ) ( x ) , F q , q )( r , r ) ( y )) , i. e. F p , p )( r , r ) ( x + y ) ≥ T ( F p , p )( q , q ) ( x ) , F q , q )( r , r ) ( y )) . Now we claim the following statements the proofs each of them consists just directlyverification.
Theorem 1
Let ( S , τ ) and ( S , τ ) be g -topological spaces of type V . Then τ × τ is a g -topology of type V on S × S . Theorem 2
Let ( S , τ D ) and ( S , τ D ) be g -topological spaces of type V D . Then τ D × τ D is a g -topology of type V D on S × S . Theorem 3
Let ( S , τ α ) and ( S , τ α ) be g -topological spaces of type V α . Then τ α × τ α is a g -topology of type V α on S × S . Theorem 4
Let ( S , τ e ) and ( S , τ e ) be g -´ecart g -topological spaces. Then τ e × τ e is a g -´ecart g -topology on S × S . Theorem 5
Let ( S , τ e ) and ( S , τ e ) be R - g -topological spaces. Then τ e × τ e is a R - g -topology on S × S . Remark 2
Note that the Cartesian product of topologies must not be a topology, i. e.usual topology does not close under product. Unlike usual topology, g -topologies areclosed with respect to product. 8 eferenceseferences