Hindawi Publishing Corporation Journal of Function Spaces Volume 2014, Article ID 753969, 6 pages http://dx.doi.org/10.1155/2014/753969 Research Article On Continuity of Functions between Vector Metric Spaces Cüneyt Çevik Department of Mathematics, Faculty of Science, Gazi University, Teknikokullar, 06500 Ankara, Turkey Correspondence should be addressed to Cuneyt¨ C¸ evik; [email protected] Received 17 August 2014; Accepted 8 October 2014; Published 12 November 2014 Academic Editor: Han Ju Lee Copyright © 2014 Cuneyt¨ C¸ evik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce vectorial and topological continuities for functions defined on vector metric spaces and illustrate spaces of such functions. Also, we describe some fundamental classes of vector valued functions and extension theorems. 1. Introduction and Preliminaries vector space has started by Zabrejko in [8]. The distance map in the sense of Zabrejko takes values from an ordered vector Let be a Riesz space. If every nonempty bounded above space. We use the structure of lattice with the vector metrics (countable) subset of has a supremum, then is called having values in Riesz spaces; then we have new results as Dedekind (-) complete.Asubsetin is called order bounded mentioned above. This paper is on some concepts and related if it is bounded both above and below. We write ↓if () results about continuity in vector metric spaces. is a decreasing sequence in such that inf =.TheRiesz −1 space is said to be Archimedean if ↓0holds for every Definition 1. Let be a nonempty set and let be a Riesz :×→ ∈+.Asequence() is said to be -convergent (or order space. The function is said to be a vector convergent)to if there is a sequence () in such that ↓0 metric (or -metric) if it satisfies the following properties: and |−| ≤ for all ,where|| = ∨(−) for any ∈.We (vm1) (, ) =0 if and only if =, will denote this order convergence by → .Furthermore,a sequence () is said to be -Cauchy if there exists a sequence (vm2) (, ) ≤ (, ) +(,) for all , ,. ∈ () in such that ↓0and | −+|≤ for all and . The Riesz space is said to be -Cauchy complete if every Also the triple (, , ) is said to be vector metric space. -Cauchy sequence is -convergent. (, , ) An operator :between → two Riesz spaces is Definition 2. Let be a vector metric space. positive if () ≥0 for all ≥0.Theoperator is said ( ) to be order bounded if it maps order bounded subsets of (a) A sequence in is vectorially convergent (or is ∈ to order bounded subsets of .Theoperator is called - -convergent) to some ,ifthereisasequence () in such that ↓0and (,)≤ for all . order continuous if →0 in implies () →0 in , .Every-order continuous operator is order bounded. If We will denote this vectorial convergence by → (∨)= ()∨()for all , ,theoperator ∈ is . a lattice homomorphism. For further information about Riesz (b) A sequence () in is called -Cauchy sequence spaces and the operators on Riesz spaces, we refer to [1, 2]. whenever there exists a sequence () in such that In [3], a vector metric space is defined with a distance map ↓0and (,+)≤ for all and . having values in a Riesz space, and some results in metric space theory are generalized to vector metric space theory. (c) The vector metric space is called -complete if each Some fixed point theorems in vector metric spaces are given -Cauchy sequence in is -convergenttoalimitin in [3–7].Actually,thestudyofmetricspaceshavingvalueona . 2 Journal of Function Spaces One of the main goals of this paper is to demonstrate the Corollary 5. For a function :between → two vector properties of functions on vector metric spaces in a context metric spaces (, , ) and (,,) the following statements more general than the continuity in metric analysis. Hence, hold. the properties of Riesz spaces will be the tools for the study of (a) If is Dedekind -complete and is vectorially contin- continuity of vector valued functions. The result we get from (( ), ()) ↓0 ( ,)↓0 here is continuity in general sense, an order property rather uous, then whenever . than a topological feature. (b) If is Dedekind -complete and ((), ()) ↓ In Section 2, we consider two types of continuity on vec- 0 whenever (,) ↓, 0 then the function is tor metric spaces. This approach distinguishes continuities vectorially continuous. vectorially and topologically. Moreover, vectorial continuity (c) Suppose that and are Dedekind -complete. Then, examples are given and the relationship between vectorial the function is vectorially continuous if and only if continuity of a function and its graph is demonstrated. In ((), ()) ↓0 whenever (,)↓0. Section 3, equivalent vector metrics, vectorial isometry, vec- , torial homeomorphism definitions, and examples related to ( ,) ↓ 0 → these concepts are given. In Section 4,uniformcontinuityis Proof. (a) If ,then .Bythevectorial continuity of the function ,thereisasequence() in discussed and some extension theorems for functions defined ↓0 (( ), ()) ≤ on vector metric spaces are given. Finally, in Section 5, such that and for all .Since is Dedekind -complete, ((), ()) ↓0 holds. a uniform limit theorem on a vector metric space is given, , and the structure of vectorial continuous function spaces is (b) Let →in . Then there is a sequence () in demonstrated. such that ↓0and (,) ≤ for all .Since is Dedekind -complete, (,)↓ 0holds. By the hypothesis, , 2. Topological and Vectorial Continuity ((), ()),andso ↓0 () → () in . (c) Proof is a consequence of (a) and (b). We now introduce two types of continuity in vector metric , spaces. For two elements and in a Riesz space, we write (,,) → < ≤ ≠ Example 6. Let be a vector metric space. If to indicate that but . , and →,then(,) → (,;thatis,thevector ) (, , ) (,, ) 2 Definition 3. Let and be vector metric metric map from to is vectorially continuous. Here, ∈ 2 ̃ spaces, and let . is equipped with the -valued vector metric defined (,̃ ) = ( , )+(, ) =(, ), = (a) A function :is → said to be topologically as 1 2 1 2 for all 1 1 ( , )∈2 continuous at if for every >0in there exists 2 2 ,and is equipped with the absolute valued |⋅| some in such that ((), ()) < whenever vector metric . ∈ (, ) < and .Thefunction is said (, , ) to be topologically continuous if it is topologically We recall that a subset of a vector metric space is called -closed (or vector closed) whenever ()⊆and continuous at each point of . , →imply ∈. (b) A function :is → said to be vectorially , , continuous at if →in implies () → Theorem 7. Let (, , ) and (,,)be vector metric spaces. () in .Thefunction is said to be vectorially If a function :is → vectorially continuous, then for −1 continuous if it is vectorially continuous at each point every -closed subset of the set () is -closed in . of . −1 Proof. For any ∈ (), there exists a sequence () in Theorem 4. (, , ) (,,) , Let and be vector metric spaces −1() → : → such that . Since the function is vectorially where is Archimedean. If a function is , topologically continuous, then is vectorially continuous. continuous, () → ().Buttheset is -closed, so ∈ −1 −1 ().Then,theset () is -closed. , Proof. Suppose that →. Then there exists a sequence If and are two Riesz spaces, then ×is also a Riesz () in such that ↓0and (,)≤ for all .Let be space with coordinatewise ordering defined as any nonzero positive element in . By topological continuity =((1/); ) of at , there exist elements in such that (1,1)≤(2,2)⇐⇒1 ≤2,1 ≤2 (1) (, )< implies ((), ( )) < (1/) for all .Then = ∨ ( ,) ≤ there exist elements in such that for all (1,1), (2,2)∈×. The Riesz space ×is < implies ((), ()) < (1/) for all .Since is a vector metric space, equipped with the biabsolute valued , vector metric |⋅|defined as Archimedean, (1/).Hence, ↓0 () → (). |−| = (1 −2 , 1 −2) (2) Vectorially continuous functions have a number of nice characterizations. for all =(1,1), =(2,2)∈×. Journal of Function Spaces 3 Let and be two vector metrics on which are - function ℎ from ×to defined by ℎ(, ) = |() −()| valued and -valued, respectively. Then the map defined for all ∈, ∈is vectorially continuous with the ×- as valued product vector metric and the absolute valued vector metric |⋅|. (,) = ( (,) ,(,)) (3) (c) If : (, , ) → (, ,) and : (,,) → , ∈ × (,, ) are vectorially continuous functions, then the func- for all is an -valued vector metric on .Wewill ℎ × × ℎ(, ) = ((), ()) call double vector metric. tion from to defined by for all ∈, ∈is vectorially continuous with the ×- × Example 8. Let (,,) and (, , ) be vector metric valued and -valued product vector metrics. spaces. We have the next proposition for any product vector (i) Suppose that :and → :are → metric. vectorially continuous functions. Then the function ℎ from to ×defined by ℎ() = ((), ()) for all Proposition 12. Let ()=(,) be a sequence in ( × ∈ ,× is vectorially continuous with the double vector , , ×) and let = (,)∈.Then, × → metric and the biabsolute valued vector metric |⋅|. , , ifandonlyif →and →. (ii) Let :×→and :×→be the ℎ: → × projection maps. Any function can be Now let us give relevance between vectorial continuity of ℎ() = ((), ()) ∈ = written as for all where a function and closeness of its graph. ℎ and =ℎ.Ifℎ is vectorially continuous, so are and since and are vectorially continuous. Corollary 13. Let (,,) and