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A Cardinal Function on the Category of Metric Spaces 1 International Journal of Contemporary Mathematical Sciences Vol. 9, 2014, no. 15, 703 - 713 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijcms.2014.4442 A Cardinal Function on the Category of Metric Spaces Fatemah Ayatollah Zadeh Shirazi Faculty of Mathematics, Statistics and Computer Science, College of Science University of Tehran, Enghelab Ave., Tehran, Iran Zakieh Farabi Khanghahi Department of Mathematics, Faculty of Mathematical Sciences University of Mazandaran, Babolsar, Iran Copyright c 2014 Fatemah Ayatollah Zadeh Shirazi and Zakieh Farabi Khanghahi. 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. Abstract In the following text in the metric spaces category we introduce a topologically invariant cardinal function, Θ (clearly Θ is a tool to classify metric spaces). For this aim in metric space X we consider cone metrics (X; H; P; ρ) such that H is a Hilbert space, image of ρ has nonempty interior, and this cone metric induces the original metric topology on X. We prove that for all sufficiently large cardinal numbers α, there exists a metric space (X; d) with Θ(X; d) = α. Mathematics Subject Classification: 54E35, 46B99 Keywords: cone, cone metric order, Fp−cone metric order, Hilbert space, Hilbert-cone metric order, metric space, solid cone 1 Introduction As it has been mentioned in several texts, cone metric spaces in the following form has been introduced for the first time in [9] as a generalization of metric 704 F. Ayatollah Zadeh Shirazi and Z. Farabi Khanghahi spaces (e.g. see [8]). Several papers has been published regarding the matter since 2007. A large number of these papers deal with fixed point theorems (e.g. see [9], [1], [6]), however there are texts deal with the other properties (e.g., [2]) amongst these texts some authors try to study metrizability or interaction between metric spaces and cone metric spaces ([5], [3], [4]). In this text R denotes the set of all real numbers, and N = f1; 2;:::g denotes the set of all natural numbers. Note. All vector spaces assumed in this text are nonzero real vector spaces. In (real) norm vector space (E; k k) we say P(⊆ E) is a cone if: • P is a closed nonempty subset of E, • for all x; y 2 P and λ, µ ≥ 0 we have λx + µy 2 P, • P \ −P = f0g, and in addition it is a solid cone in E, if P◦ 6= ;. Suppose P is a cone in norm vector space E. For all x; y 2 E we say x ≤P y if y − x 2 P. Obviously in this way (E; ≤P) is a partial ordered set. For all x; y 2 E we say x <P y if x ≤P y and x 6= y, moreover we say x P y or simply x y if y − x 2 P◦. We say (X; E; P; d) is a cone metric space if P is a cone in E and d : X ×X ! P for all a; b; c 2 X satisfies the following conditions: • d(a; b) = 0 if and only if a = b, • d(a; b) = d(b; a), • d(a; b) ≤P d(a; c) + d(c; b). In the cone metric space (X; E; P; d) for " 0 and x 2 E let Bd(x; ") or simply B(x; ") is fz 2 E : d(z; x) "g. It is easy to see that fB(x; "): x 2 E;" 0g is a topological basis on X. We consider cone metric space (X; E; P; d) under topology generated by the above basis on X. It is well-known that for every (real) Hilbert space H there exists a nonzero cardinal number α such that H and `2(α) are isomorphic (as Hilbert spaces), where for nonempty set Γ we have: 8 9 2 < Γ X 2 = ` (Γ) = (xλ)λ2Γ 2 R : jxλj < 1 : λ2Γ ; X equipped with inner product < (xλ)λ2Γ; (yλ)λ2Γ >:= xλyλ and therefore λ2Γ 1 0 1 2 X 2 2 norm k(xλ)λ2Γk = @ jxλj A (for (xλ)λ2Γ; (yλ)λ2Γ 2 ` (Γ)). λ2Γ A cardinal function on the category of metric spaces 705 We recall that for all nonzero cardinal number α, we have α = fβ 2 ON : β < αg where ON is the class of all ordinal numbers (by CN we mean the class of all cardinal numbers which is a proper subclass of ON). We denote the least infinite cardinal number with @0 or ! and the cardinality of R with c. Also it is well-known that `2(n) can be considered as Rn with Euclidean norm for 1 ≤ n < !. 2 First steps In this section we get ready to define Hilbert-cone metric order of a metric space (X; d). Lemma 2.1 In norm vector space (F; k k) if V is a nonvoid open subset of F, then card(V ) = card(F) = max(c; α), where α = dimR(F) (α is the cardinality of a Hamel basis of F over R). Proof. For r > 0 let Ur = fx 2 F : kxk < rg. For s > 0, card(Ur) = card(Us), s S since ' : Ur ! Us with '(x) = r x is bijective. On the other hand F = fUn : n 2 Ng, which leads to (since for all n 2 N we have card(U1) = card(Un)): card(U1) ≤ card(F) ≤ @0card(U1) : Using @0 < card(R) ≤ card(F) ≤ @0card(U1) we have @0card(U1) = card(U1). By card(U1) ≤ card(F) ≤ @0card(U1) = card(U1) we have: card(F) = card(U1) : Suppose V is a nonvoid open subset of F, there exist z 2 V and r > 0 such that z + Ur ⊆ V . Since η : U1 ! V with η(x) = rz + x is injective, we have: card(F) ≥ card(V ) ≥ card(U1) = card(F) : Therefore card(V ) = card(F) = max(c; α). Theorem 2.2 In cone metric space (X; E; P; d) if d(X × X)◦ 6= ;, then: card(X) ≥ card(E) : Proof. Using Lemma 2.1, card(d(X × X)◦ = card(E). Moreover card(X × X) ≥ card(d(X × X)) ≥ card(d(X × X)◦) = card(E) and X is infinite, therefore card(X) = card(X × X) ≥ card(E). Using Theorem 2.2, in cone metric space (X; E; P; d) if card(X) < card(E), then d(X × X)◦ = ;. 706 F. Ayatollah Zadeh Shirazi and Z. Farabi Khanghahi Lemma 2.3 For α ≥ c we have card(`2(α)) = α. 2 Proof. Consider α ≥ c, for all (xθ)θ2α 2 ` (α) there exists a countable subset 2 D of α such that for all θ 2 α n D we have xθ = 0. Therefore card(` (α)) ≤ card(f(D; (xθ)θ2D): D is a countable subset of α and xθ 2 R for all α 2 Dg), 2 which leads to card(` (α)) ≤ card(f(f; (xn)n2N): f : N ! α is an injection and xn 2 R for all n 2 Ng) and: card(`2(α)) ≤ card(αN × RN) = α@0 c@0 = α therefore card(`2(α)) = α. Definition 2.4 In metric space (X; d) let Θ(X; d) := sup(f0g [ fα 2 CN : there exists cone metric space (X; `2(α); P; ρ) which induces the same topology as metric topology on (X; d) and ρ(X × X)◦ 6= ;g). Using Theorem 2.2 and Lemma 2.3, Θ(X; d)(≤ card(X)) exists. We call Θ(X; d), cone metric order of (X; d) with respect to Hilbert spaces, or sim- ply Hilbert-cone metric order of (X; d). Note. Naturally Θ gives a classification of metric spaces. Instead of classifying metric spaces regarding collection f`2(α): α 2 CN n f0gg, one may consider p classification regarding Fp = f` (α): α 2 CN n f0gg for 1 ≤ p ≤ +1, so let Θp(X; d) := sup(f0g [ fα 2 CN : there exists cone metric space (X; `p(α); P; ρ) which induces the same topology as metric topology on (X; d) and ρ(X × X)◦ 6= ;g). And call Θp(X; d), cone metric order of (X; d) with respect to Fp, or simply Fp- cone metric order of (X; d). Now one may be interested to study the relation between Θp(X; d) and Θq(X; d) for p; q ≥ 1. 3 Towards main theorem: A useful example The main aim of this section is to prove that for β ≥ c there exists metric space (Z; ) with Θ(Z; ) = β. In this section suppose α 2 CN is a nonzero cardinal number and G = (Gθ)θ2α where G0 = 1 and Gθ = 0 for θ 6= 0. Moreover set 2 P = f(xλ)λ2α 2 ` (α): x0 ≥ 0 ^ (8β 2 α (jxβj ≤ x0))g ; Q = f(xθ)θ2α 2 P : supfjxθj : θ 6= 0g < min(x0; 1)g ; A cardinal function on the category of metric spaces 707 X = `2(Q) n f0g : v v v Also for v 2 Q, suppose uv = (δθ )θ2Q with δv = 1 and δθ = 0 for θ 6= v. Now define d : X × X ! R with: 8 min(jλ − µj; 1) v 2 Q; λ, µ 2 R; x = λu ; y = µu <> v v d(x; y) = 0 x = y :> 4 otherwise and ρ : X × X ! P with: 8 d(x; y)v v 2 Q; x; y 2 Ru <> v ρ(x; y) = 0 x = y :> 4G otherwise Lemma 3.1 The set P is a solid cone in `2(α).
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