Metrization theorem for a weaker class of uniformities

Metrization theorem for a weaker class of uniformities

P. Sumati Kumari, I. Ramabhadra Sarma & J. Madhusudana Rao

Afrika Matematika ISSN 1012-9405 Afr. Mat. DOI 10.1007/s13370-015-0369-9

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Author's personal copy Afr. Mat. DOI 10.1007/s13370-015-0369-9

Metrization theorem for a weaker class of uniformities P. Sumati Kumari1 · I. Ramabhadra Sarma2 · J. Madhusudana Rao3

Received: 23 October 2014 / Accepted: 17 July 2015 © African Mathematical Union and Springer-Verlag Berlin Heidelberg 2015

Abstract Pseudodislocated uniformities are defined and a Metrization theorem is derived in terms of a weaker class of metric spaces. Keywords

Pseudodislocated uniformities · Metrization theorem

Mathematics Subject Classification

54E35

1 Introduction Uniform spaces were introduced in 1937 by Weil [1] (by means of entourages; the definition of uniform spaces by means of uniform coverings was given in 1940, see [2]). Initially, uniform spaces were used as tools for the study of the topologies. A metric on a metrizable space was often used for the study of the topological properties of the space. This note deals with extending the concept of Alexandroffs uniformization process to certain weaker forms of metric spaces. The metrization theorem establishes that a uniform space is pseudometrizable if and only if the uniformity has a countable basis. The authors attempt to extend this notion for weaker forms of metric spaces along the lines of Alexandroffs results in isolating a specific class namely pseudodislocatedmetric spaces for which this extension is possible. Definition 1.1 [3] If X is a set, d : X × X −→ R + is a pseudometric on X if 1. d(x, x) = 0 for every x ∈ X 2. d(x, x) = d(y, x) for every x, y ∈ X and

B

P. Sumati Kumari [email protected]

1

Department of Mathematics, K.L. University, Vaddeswaram, Guntur, Andhra Pradesh, India

2

7th Line, A.T. Agraharam, Guntur 522004, Andhra Pradesh, India

3

Department of Mathematics, Vijaya College of Engineering, Khammam, Telangana, India

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3. d(x, z) ≤ d(x, y) + d(y, z) for every x, y, z ∈ X If d is a pseudometric on X then the collection of open balls {Br (x)/x ∈ X, r > 0}. This topology is said to be generated (induced) by d on X . A topological space (X, ) is called pseudometrizable if  is generated by a pseudometric. It is well known that a pseudometric gives rise to a uniformity with base {Br (x)/r > 0} where Br (x) = {(x, y)/d(x, y) < r }. Definition 1.2 [3] Let X be a nonempty set. A nonempty collection U of subsets of X × X is a uniformity for X if (U1 ) the diagonal  = x = {(x, x)/x ∈ X } ⊆ U ∀U ∈ U (U2 ) U ∈ U ⇒ U −1 = {(y, x)/(x, y) ∈ U } ∈ U (U3 ) U ∈ U ⇒ there exist V ∈ U such that V ◦ V ⊂ U (U4 ) U ∈ U , V ∈ U ⇒ U ∩ V ∈ U and (U5 ) U ∈ U and U ⊆ V ⊆ X × X ⇒ V ∈ U . If U is a uniformity for X , the pair (X, U ) is called a uniform space. It is well known that if d is a pseudometric on a set X then the collection Ud = {U ⊂ X × X/Br (x) ⊆ U for some r > 0} is a uniformity and the collection B = {Br (x)/r > 0} is a base for the uniformity U . It is proved in Kelly’s “General Topology” [3] that a uniform space is pseudometrizable if and only if its uniformity has a countable base. In this note we ask a similar question for a weak form of pseudometric space, which we call pseudodislocated metric space and derive an answer in the form of d-metrization theorem analogous to the metrization theorem.

2 Metrization theorem for pseudodislocated metric space Now, we introduce the following definition. Definition 2.1 A pseudodislocated metric space is a pair (X, d) where X is a nonempty set and d is a pseudodislocated metric on X . i.e. d : X × X −→ R + satisfies (d1 ) d(x, y) = d(y, x) for every x, y ∈ X and (d2 ) d(x, y) ≤ d(x, z) + d(z, y) for every x, y, z ∈ X . Example Let X = R + and define d : X × X → R + be the function defined as d(x, y) = x +y for all x, y ∈ X. Then (X, d) is a pseudodislocated metric space. A pseudodislocated metric space (X, d) is dislocated metric space if (d3 ) d(x, y) = 0 ⇒ x = y and a pseudometric space (d4 ) d(x, x) = 0∀x ∈ X . Topological aspects and fixed point theorems on dislocated metric space can be found in [4–12]. It is clear that a pseudodislocated metric space is a metric space if (d3 ) and (d4 ) both hold. Definition 2.2 A pseudodislocated uniformity U A for a set X associated with a subset A of X is a nonvoid family of subsets of X × X which satisfies U2 , U3 , U4 , U5 of Definition 1.2

together with the following modified version of U1 of U1 :

U1 Every member of U A contains  A = {(x, x)/x ∈ A}.

If U A is a pseudodislocated uniformity for X , the pair (X, U A ) is called a dislocated uniform space (simply d-uniform space). Remark If A = X , the pseudodislocated uniformity is a uniformity.

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Proposition 2.3 Let (X, d) be a pseudodislocated metric space and A = {x ∈ X/d(x, x) = 0} the collection U A = {U/Br (x) ⊆ U for some r > 0}, where Br (x) is a pseudodislocated uniformity on X associated with A. Proof Clearly  A ⊆ Br (x) ∀r > 0. Also Br−1 (x) = {(y, x)/(x, y) < r } = Br (x) ∀r > 0. If r > 0, B r2 (x) ◦ B r2 (x) ⊆ Br (x)∀r > 0 and Br (x) ⊆ Bs (x) if 0 < r < s. Thus  A ⊆ U ∀U ∈ U , that U2 through U5 hold for U is a consequence of the above observations. Hence U is a pseudodislocated uniformity associated with A.   Theorem 2.4 Let X be a nonempty set, A ⊆ X and {Un /n ≥ 0} be a contracting sequence of subsets of X × X such that (i) (ii) (iii) (vi)

U0 = X × X  A ⊆ Un ∀n Un+1 ◦ Un+1 ⊆ Un ∀n Un+1 ◦ Un+1 ◦ Un+1 ⊆ Un ∀n. Then

(a) there is a nonnegative real valued function d on X × X such that d(x, y) + d(y, z) ≥ d(x, z) ∀ x, y, z ∈ X (b) Un ⊆ {(x, y)/d(x, y) < 21n } ⊆ Un−1 ∀n ≥ 1 and (c) d(x, x) = 0 ∀x ∈ A. Further (d) if each Un is symmetric then d(x, y) = d(y, x) ∀ x, y, ∈ X . Our proof is actually adapted from that of the metrization lemma in Kelly’s “General Topology” [3]. In the process we have noticed that case (ii) of our proof somehow missed the attention in the Metrization Lemma. Proof Define f : X × X → [0, ∞) by  0, f (x, y) =

1 2n ,

if (x, y) ∈ Un ∀n if (x, y) ∈ Un−1 − Un

(1)

n d(x, y) = I n f { i=0 f (xi , xi+1 )/n > 0, x = x0 , . . . , xn , xn+1 = y, xi ∈ X, 0 ≤ i ≤ n}. It is clear that 0 ≤ d(x, y) ≤ f (x, y) for all x, y and d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z. This proves (a). The proof of the first inclusion in (b) is the same as that in [3]. To prove the second inclusion, we first prove that for every choice of x0 , . . . , xn , xn+1 . f (x0 , xn+1 ) ≤ 2

n 

f (xi , xi+1 )

(2)

i=0

when n = 0, f (x0 , x1 ) < 2 f (x0 , x1 ). Let us now make k the induction assumption that if x = x0 , y = xk+1 f (x, y) = f (x0 , xk+1 ) ≤ 2 i=0 f (xi , xi+1 ). For all x, y ∈ X and any elements x = x0 , . . . , xn , xn+1 = y of X write a = n f (x , x ). i i+1 i=0

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We now consider two cases: case (i) f (x0 , x1 ) = f (x, x1 ) ≤ a2 . In this case the proof goes along the lines of [1], hence is omitted. case (ii) f (x0 , x1 ) = f (x, x1 ) ≥ a2 then f (x1 , x2 ) + · · · + f (xn , xn+1 ) < a2 so that by induction hypothesis,f (x1 , xn+1 ) < 2( a2 ) = a. n Also f (x0 , x1 ) ≤ i=0 f (xi , xi+1 ) = a. Since Um ◦ Um ⊆ Um−1 it follows that (x0 , xn+1 ) ∈ Um−1 . 1 To complete nthe proof of (b) using (2), we note that if d(x, y) = d(x 0 , xn+1 ) < 2n then f (x, y) ≤ 2 i=0 f (xi , xi+1 ) ∀n. n 1 f (xi , xi+1 ) = 2d(x0 , xn+1 ) = 2d(x, y) < 2n−1 ⇒ f (x, y) ≤ 2 i=0 ⇒ (x, y) ∈ Un−1 This completes the proof of (b). (c) (x, y) ∈  A ⇒ (x, y) ∈ Un ∀ n ⇒ f (x, y) = 0 ⇒ d(x, y) = 0 (d) If each Un is symmetric, f (x, y) = f (y, x) for every x, y ∈ X . This completes the proof.

 

Corollary 2.5 Let {Un /n ≥ 0} be a sequence of subsets of X × X such that (i) U0 = X × X (ii)  ⊆ U0 ∀n ≥ 0 (iii) Un+1 ◦ Un+1 ◦ Un+1 ⊆ Un ∀n ≥ 0. Then there is a nonnegative real valued function d on X × X such that (a) d(x, y) + d(y, z) ≥ d(x, z) ∀ x, y, z ∈ X and (b) Un ⊆ {(x, y)/d(x, y) < 2−n } ⊆ Un−1 . If each Un is symmetric there is a pseudodislocated metric d on X satisfying condition (b). Proof Let A = X in Proposition 2.3. Then  A =  = diagonal o f X × X . That Un s are contracting follows from (3), since Un+1 =  X ◦  X ◦ Un+1 ⊆ Un+1 ◦ Un+1 ◦ Un+1 ⊆ Un . Also Un+1 ◦ Un+1 =  X ◦ Un+1 ◦ Un+1 ⊆ Un+1 ◦ Un+1 ◦ Un+1 ⊆ Un . Hence Theorem 2.4 is applicable.   Definition 2.6 If U A is a pseudodislocated uniformity for X associated with A ⊆ X, U ⊆ U A is said to be a basis for U A if for every V ∈ U A there exists U ∈ U such that U ⊆ V . Definition 2.6(a): The power set P (X × X ) of a set X × X is the set of all subsets of X × X.

Lemma 2.7 A nonvoid family U of subsets of X × X is a base for some pseudodislocated uniformity U A for X associated with A ⊂ X if and only if (i) (ii) (iii) (iv)

Each number of U contains  A If U ∈ U , then U −1 contains a number of U If U ∈ U , then V ◦ V ⊂ U for some V ∈ U , and The intersection of two members of U contains a member of U .

Proof If U is a base for a pseudodislocated uniformity U A for some A ⊂ X then clearly (i) through (iv) hold. Conversely suppose U ⊂ P (X × X ) satisfies (i) through (iv) above.

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(a) Clearly  A ⊆ V ∀V ∈ U A (b) V ∈ U A , U ∈ U and U ⊆ V ⇒ U −1 ∈ U and U −1 ⊆ V −1 . So V −1 ∈ U A (c) V ∈ U A , U1 ∈ U and U1 ⊆ V implies there exists U ∈ U  U ◦ U ⊆ U1 . Hence U ◦ U ⊆ V (d) V1 ∈ U A , V2 ∈ U A there exists U1 , U2 ∈ U such that Ui ⊆ Vi , i = 1, 2. Since U1 , U2 ∈ U there exists U ∈ U such that U ⊆ U1 ∩ U2 . ⇒ U ⊆ V1 ∩ V2 ⇒ V1 ∩ V2 ∈ U A . (e) Finally if U ∈ U and U ⊆ V , there exists U1 ∈ U  U1 ⊆ U ⊆ V . So V ∈ U A .  

Hence U A is a pseudodislocated uniformity associated with A.

Lemma 2.8 If a pseudodislocated uniformity U A for X associated with A ⊂ X has countable base {Vn /n ≥ 0} then there exists a base {Un /n ≥ 0} for U A such that U A is symmetric, Un ◦ Un ⊂ Un−1 and Un ◦ Un ◦ Un ⊂ Un−1 for every n ≥ 0.

Proof We may assume that V0 = X × X and Vn ⊂ Vn−1 for n ≥ 0, since {Vn /n ≥ 0} where

Vn = X × X and Vn = Vn−1 ∩ Vn is also a basis for U . We may also assume that each Vn is symmetric as we may replace Vn by Vn ∩ Vn−1 ∀n. Clearly, V1 ◦ V1 ◦ V1 ⊂ V0 . Choose n 1  Vn 1 ◦ Vn 1 ⊆ V1 and m 1 ≥ n 1 such that Vm 1 ◦ Vm 1 ◦ Vn 1 ⊆ V1 then Vm 1 ◦ Vm 1 ◦ Vm 1 ⊆ Vn 1 ◦ Vm 1 ⊆ Vn 1 ◦ Vn 1 ⊆ V1 . Write U0 = V0 , . . . , U1 = V1 and U2 = Vm i . Assume that U0 , U1 , U2 , . . . , Un are chosen so that Uk ⊆ Vk for 0 ≤ k ≤ n, Uk ◦ Uk ⊆ Uk−1 and Un ◦ Un ◦ Un ⊆ Un−1 for 1 ≤ k ≤ n. As above there exist n k , m k ≥ n k such that Vn k ◦ Vn k ⊆ Un and Vm k ◦ Vm k ◦ Vn k ⊆ Vn then Vm k ◦ Vm k ◦ Vm k ⊆ Vn ◦ Vn k ⊆ Un . Write Un+1 = Vm k . Then Un+1 ◦ Un+1 ◦ Un+1 ⊆ Uk and Un+1 ◦ Un+1 ⊆ Un . By induction we get a countable collection {Un /n ≥ 0}  Un ⊆ Vn , Un+1 ◦ Un+1 ⊆ Vn and Un+1 ◦ Un+1 ◦ Un+1 ⊆ Un ∀n. Clearly {Un /n ≥ 0} is a basis for U A .  

3 Metrization theorem for pseudodislocated metric space Theorem 3.1 A dislocated uniform space (X, U A ) is pseudodislocated metrizable if and only if U A has a countable base. Proof It is clear that a dislocated uniform space (X, U A ) is pseudodislocated metrizable if U A has a countable base. On the other hand, If U A has a countable base then by Lemma 2.8, U A has a countable base satisfying the conditions of Theorem 2.4. Hence there is a pseudodislocated metric d satisfying the conclusion of Theorem 2.4. Hence (X, U A ) is pseudometrizable. This completes the proof of the theorem.  

References 1. Weil, A.: Sur les espaces structure uniforme et sur la topologie generale. Hermann, Paris (1938). [Zbl 0019.18604, Paris: Hermann and Cie. 40 p. (1938)] 2. Tukey, J.W.: Convergence and Uniformity in Topology. Princeton Univ. Press, Princeton (1940) 3. Kelley, J.L.: General Topology. D. Van Nostrand Company Inc., New York (1960)

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Author's personal copy P. S. Kumari et al. 4. Sarma, I.R., Kumari, P.S.: On dislocated metric spaces. Int J Math Arch 3(1), 72–77 (2012) 5. Kumari, P.S., Kumar, V.V., Sarma, I.R.: New version for Hardy and Rogers type mapping in dislocated metric space. Int J Basic Appl Sci 1(4), 609–617 (2012) 6. Kumari et al.: Common fixed point theorems on weakly compatible maps on dislocated metric spaces. Math Sci 6, 71 (2012). doi:10.1186/2251-7456-6-71 7. Sumati Kumari, P., Ramabhadra Sarma, I., Madhusudana Rao, J., Panthi, D. : Completion of a Dislocated Metric Space. Abstr Appl Anal 2015, 5 (2015) doi:10.1155/2015/460893 8. Kumari, P.S., Zoto, K., Panthi, D.: d-Neighborhood system and generalized F-contraction in dislocated metric space. SpringerPlus 4.1 1–10 (2015) 9. Karapinar E., Salimi, P.: Dislocated metric space to metric spaces with some fixed point theorems. Fixed Point Theory Appl 2013, 222 (2013) 10. Zoto, K., Hoxha, E., Kumari, P.S.: Some fixed point theorems and cyclic contractions in dislocated and dislocated quasimetric spaces. Turkish J Anal Number Theory 2(2), 37–41 (2014). doi:10.12691/ tjant-2-2-2 11. Kumari, P.S., Ramana, Ch.V., Zoto, K., Panthi, D.: Fixed point theorems and generalizations of dislocated metric spaces. Indian J Sci Technol 8(S3), 154–158 (2015) 12. Kumari, P.S.: On dislocated quasi metrics. J Adv Stud Topol 3(2), 66–74 (2012)

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