Uniform Convergence and Uniform Continuity in Generalized Metric Spaces


 Johnathan Franklin
 1 years ago
 Views:
Transcription
1 Int. Journal of Math. Analysis, Vol. 5, 2011, no. 6, Uniform Convergence and Uniform Continuity in Generalized Metric Spaces Abdul Mohamad Department of Mathematics and Statistics Sultan Qaboos University, Oman Abstract The class of generalized metric spaces is introduced and studied in [9]. In this paper, the concepts of uniform convergence, uniform continuity and BolzanoWeierstrass property in this class are investigated. We show that many theorems in classic metric spaces about these concepts are still valid for this class of spaces. Mathematics Subject Classification: 54A05, 54A20, 54E70, 54H25 Keywords: Generalized metric space; Probabilistic metric space; Uniform Convergence; Uniform Continuity 1 Introduction In the paper [9] we introduced the class of generalized metric spaces. These spaces simultaneously generalize standard metric spaces, probabilistic metric spaces and fuzzy metric spaces. We show that every generalized metric space is, naturally, a uniform space. Thus we can use standard topological techniques to study, for example, probabilistic metric spaces. We illustrate this by proving a Fixed Point Theorem for uniform spaces, and interpret this result in the context of probabilistic metric spaces. A uniform structure (or uniformity) on a set X is a non empty set U of subsets of X X which satisfies the following axioms: 1. Every subset of X X which contains a set of U belongs to U. 2. Every finite intersection of sets of U belongs to U.
2 286 A. Mohamad 3. Every set of U is a reflexive relation on X (i.e. contains the diagonal). 4. If V belongs to U, then V = {(y, x) :(x, y) V } belongs to U. 5. If V belongs to U, then exists V in U such that, whenever (x, y), (y, z) V, then (x, z) V (i.e. V V V ). The sets of U are called entourages or vicinities. The set X together with the uniform structure U is called a uniform space. Recall that a uniformity U on a set X has the Lebesgue property provided that for each open cover G of X there is U Usuch that {U(x) :x X} refines G, and U is called equinormal if for each pair of disjoint nonempty closed subsets A and B of X there is U Usuch that U(A) B =. A metric d on X has the Lebesgue property provided that the uniformity U d, induced by d, has the Lebesgue property and d is equinormal provided that U d have it. 2 Generalized Metric Spaces A range set (G, ) is a partially ordered set with minimum element 0 so that G \{0} is downwards directed. A triangle function is a binary operation τ : G G G which is commutative, associative, has 0 as it s identity, is increasing in each component, and is continuous at 0 for all a>0 there is a b>0 such that τ(p, q) <afor all p, q < b. (Note that if we topologise G so that 0 has neighbourhoods of the form N a = {b : b<a}, where a>0, then the condition τ continuous at 0 is indeed equivalent to τ being continuous at 0.) Definition 1 Let S be a nonempty let (G, ) be a range set, and let τ be a triangle function over G. A function F : S S G is a generalized quasimetric [7] if the following conditions are satisfied for all p, q, r S. GQM1: F(p, p) =0; GQM2: if F(p, q) =0and F(q, p) =0, then p = q; GQM3: F(p, q) τ(f(p, r), F(r, q)) If F is a generalized quasimetric then the triple (S, F,τ) is called a generalized quasimetric space (abbreviated GQM space). A generalized quasimetric F is called a generalized metric if it satisfies the symmetrycondition. GQM4: F(p, q) =F(q, p) for all p, q in S. In the latter case, (S, F,τ) is a generalized metric spaces (abbreviated GM space).
3 Uniform convergence and uniform continuity 287 We often write F pq in place of F(p, q). Standard Metric Spaces If we take G to be the nonnegative real numbers, R +, with the standard order, and τ to be addition, then the generalized (quasi )metric spaces over G and τ are precisely the standard (quasi)metric spaces. Probabilistic Metric Spaces In [6], Menger introduced and studied the class of probabilistic metric spaces. A probabilistic metric space is a generalized metric space appropriate to the study of situations in which distances are measured in terms of distribution functions rather than nonnegative real numbers. Let + be the class of all distance distribution functions. A function F :[0, ) [0, 1] is called a distance distribution function if F satisfies the following conditions: 1. F (0)=0 and lim F (x) =1 x 2. F is nondecreasing 3. F is left continuous on [0, ). The class + is equipped with the modified Levy distance. The modified Levy distance [11] is the function d L defined on + + by d L (F, G) = inf{h >0:G(x) F (x + h) +h and F (x) G(x + h) +h, x (0, 1 )}. h The distance d L is a metric on +. The class + is ordered by the relation F G F (x) G(x), x 0. Let ɛ 0 be the element of + defined by { 0, if x =0 ɛ 0 = 1, if x>0 A probabilistic (quasi)metric space is precisely a generalized (quasi)metric space where the range set (G, )is( +, ) with zero element ɛ 0, and a triangle function. We abbreviate probabilistic (quasi)metric space to P(Q)M. In [5], P.S. Marcus gives an example of PQM space based on stationary Markov chains which is not a PM space. An excellent treatment of PM spaces is given in [11]. Fuzzy Metric Spaces Let S be a nonempty set. Then fuzzy metric spaces (see [2]) are precisely those generalized metric spaces where the range set G is the set of all fuzzy sets on S, and the triangle function comes from a continuous tnorm. A binary operation :[0, 1] [0, 1] [0, 1] is a continuous tnorm if satisfies the conditions:
4 288 A. Mohamad 1. is associative and commutative. 2. a 1=a for every a [0, 1] 3. a b c d whenever a c and b d, a, b, c, d [0, 1]. A tnorm gives rise to a triangle function as follows: F (F, G)(t) =F (t) G(t). Example 2 The four basic examples of tnorms are: 1. The Lukasierviez tnorm: L: I I I,a L b = max{a + b 1, 0}. 2. The product tnorm: a P b = ab. 3. The minimum tnorm: a M b = min{a, b}. 4. The weakest tnorm, the drastic product: { min{a, b}, if max(a, b) =1 D (a, b) = 0, otherwise Using pointwise ordering, we have the inequalities D < L < P < M. 3 Uniform Topologies of GM Spaces Theorem 3 Let (S, F,τ) be a GM space. The filter U = {U S S : U U(a) for some 0 <a G}, where U(a) ={(p, q) S S : F pq <a}, is a uniformity. Proof. We check off the defining properties of a uniformity: 1. Each U in U contains the diagonal because F pp = 0 for all p. 2. The family U is upwards closed by definition (the U(a) s form a basis). 3. It is clear that if U belongs to U, then U 1 = {(q, p) :(p, q) U} belongs to U because F(p, q) =F(q, p) for all p, q in S. 4. The family U is closed under finite intersections because G\{0} is downwards directed.
5 Uniform convergence and uniform continuity It remains to verify that if U is in U, then there is a V in U such that V V U. We may assume U = U(a) for some a>0. By continuity of τ at 0, there is a b>0such that if p,q <bthen τ(p,q ) <a. Let V = U(b). Take any (p, q) inv V. So there is an r such that (p, r), (r, q) U(b). Which means F pr <band F rq <b. Now we see that F pq τ(f pr, F rq ) <a in other words, (p, q) U(a), as required. Fix a GM space (S, F,τ). From the uniformity, we can obtain the topology σ associated with F which has for its neighborhood base at p S the collection {U p (a) :U U,t>0} where U p (a) ={q S :(p, q) U(a)}. Lemma 4 Let (S, F,τ) be a GM, with associated uniformity U and topology σ. 1. (S, σ) is T 0 2. (S, σ) is T 1 iff F pq =0 p = q (GQM2 ) 3. (S, σ) is metrizable iff GQM2 and (G, ) has countable cofinality (there is a countable C G \{0} such that for all a>0 there is a c C such that c<a). Proof. For 1 just note that (S, σ) being T 0 is exactly the content of axiom GQM2. Part 2 is similar. For 3 we recall that a space is metrizable iff it is T 1 and has a compatible uniformity with a countable base. A countable base for the uniformity U associated with (S, F,τ) corresponds directly, by the definition of the associated uniformity, to a countable cofinal set for (G, ). We now establish a dictionary connecting properties of generalized metric spaces and properties of the associated uniformities. Definition 5 Let (X, U) be a uniform space. 1. A sequence (x n ) n N in X is called UCauchy sequence if for each U U there exists k N such that (x r,x s ) U for all s r k. 2. A sequence (x n ) n N in X is called convergent to x X if for all U U there exists k N such that (x, x n ) U for all n k. 3. (X, U) is called Ucomplete if every UCauchy sequence is Uconvergent.
6 290 A. Mohamad Definition 6 Let (S, F,τ) be GM space. A sequence {x n } n N is called a F Cauchy sequence if for each a>0 there exists k N such that F xrxs <afor all s r k. {x n } n N is called σconvergent to x S if for every 0 <a G there exists k N such that F xxn <afor all n k. Definition 7 We say that a GM space (S, F,τ) is F complete if every F Cauchy sequence is σconvergent. It is easy to prove the following lemma. Lemma 8 Let U F be the uniformity determined by (S, F,τ). Then (S, F,τ) is Fcomplete iff (X, U) is Ucomplete. 4 Uniform Continuity Definition 9 Let (S 1, F 1,τ 1 ) and (S 2, F 2,τ 2 ) be generalized metric spaces with range sets (G 1,< 1 ) and (G 2,< 2 ) respectively. A function f : S 1 S 2 is called uniformly continuous if for every g 2 G 2, there exists g 1 G 1 such that if F 1 (p, q) <g 1 then F 2 (f(p),f(q)) <g 2. It easy to show that every uniformly continuous function is continuous. However, the converse may not be true. We shall see next that these two concepts, as in the standard metric, agree on certain kind of generalized metric spaces called compact. Theorem 10 Let (S, F,τ) and (S, F,τ ) be generalized metric spaces with range sets (G, <) and (G,< ) respectively. If the function f : S S is continuous and S is compact then f is uniform continuous. Proof. Let g G, be given, then there exists h G such that τ (h,h ) < g. Since f is continuous, for each p S, there extis h G, such that if F(p, q) <hthen F (f(p),f(q)) <h q S. But h G and therefore there extis g G such that τ(g, g) <h. From compactness of S there exist s 1,s 2,..., s n S such that S = n i=1 B(s i,g si ). Let g 0 = min{g si : i = 1, 2,..., n}. Now for any p, q S, iff(p, q) <g 0 then F(p, q) <g si. Since p S, there exists an s i such that p B(s i,g 0 ). This implies that F(p, s i ) < g si. Therefore F (f(p),f(s i )) <h. Now F(p, s i ) τ(f(p, q), F(q, s i )) τ(g si,g si ) <h si. Therefore F (f(p),f(s i )) <h. But
7 Uniform convergence and uniform continuity 291 F (f(p),f(q)) τ (F (f(p),f(s i )), F (f(s i ),f(q))) τ (h,h ) < g. completes the proof, i.e. f is uniformly continuous. This The Proof of the following theorem is similar as in classic (standard) metric spaces. Theorem 11 Let (S, F,τ) and (S, F,τ ) be generalized metric spaces. If f : S S is uniformly continuous and {x n } is Cauchy sequence in S, then {f(x n )} is also a Cauchy sequence in S. Definition 12 Suppose that X is a nonempty set and (Y,F,τ ) is a generalized metric space. A sequence {f n } of functions from X to Y is converge uniformly to a function f : X Y if for each r G, there exists n 0 N such that F (f n (x),f(x)) <rfor all n n 0 and for all x X. Definition 13 Let (X, F,τ) and (Y,F,τ ) be generalized metric spaces. A family of functions F = {f} from X to Y is called equicontinuous if for each r G, there exists r G such that if F(p, q) <rthen F (f(p),f(q)) <r, for all f F and for all p, q X. Definition 14 A real valued function f on the generalized metric space (X, F,τ) is Runiformly continuous provided that for each ɛ>0 there exists r G such that if F(x, y) <rthen f(x) f(y) <ɛ. Definition 15 A generalized metric space (X, F,τ) is called equinormal if for each pair of disjoint nonempty closed subsets A and B of (X, T F ) such that sup{f(a, b) :a A, b B} > 0. Definition 16 A generalized metric space (X, F,τ) has the Lebesgue property if for each open cover G of (X, T F ) there exist r G such that {B F (x, r) :x X} refines G. The proofs of the following lemmas and theorems are similar to those lemmas and theorems in [1] and [3]. Lemma 17 Let (X, F,τ) and (Y,F,τ ) be generalized metric spaces and let F = {f n } be an equicontinuous sequence of functions from X to the complete space Y.IfF = {f n } converges for each point of a dense subset D of X, then F = {f n } converges at each point of X and the limit function is continuous.
8 292 A. Mohamad Theorem 18 AscoliArzela Theorem Let (X, F,τ) and (Y,F,τ ) be generalized metric spaces and suppose that X is compact and Y is complete. Let A = {f} be an equicontinuous family of functions from X to Y and let F = {f n } be a sequence in A such that {f n (x)} is a compact subset of Y for each x X. Then there exists a continuous function f : X Y and a subsequence {g n } of F such that g n converges uniformly to f on X. Theorem 19 Let X be any nonempty set and (Y,d) be a metric space and let (Y,F d,τ ) be the induced generalized metric space. Then a sequence of functions {f n } from X to Y converges uniformly to a function f from X to Y with respect to the metric d iff {f n } converges uniformly to f with respect to the generalized metric F d. Theorem 20 Suppose that (X, d) and (Y,d ) are metric spaces. Let (X, F d,τ) and (Y,F d,τ ) be the corresponding induced generalized metric spaces. Then a family F of functions from X to Y is equicontinuous with respect to the metric iff F is equicontinuous with respect to the generalized metric. F d. Corollary 21 Let (X, d) and (Y,d ) be metric spaces and suppose that X is compact and Y is complete. Let A = {f} be an equicontinuous family of functions from X to Y and let F = {f n } be a sequence in A such that {f n (x)} is a compact subset of Y for each x X. Then there exists a continuous function f : X Y and a subsequence {g n } of F such that g n converges uniformly to f on X. Theorem 22 Suppose that (X, d) is a metric space and let (X, F d,τ) induced generalized metric spaces. Then X is equinormal (has Lebesgue property) with respect to the metric iff X is equinormal (has Lebesgue property) with respect to the generalized metric. Theorem 23 Let (X, F,τ) be a generalized metric space. Then following are equivalent. 1. For each generalized metric space (Y,F,τ ), any continuous mapping from (X, T F ) to (Y,T F ) is uniformly continuous as a mapping from (X, F,τ) to (Y,F,τ ). 2. Every real valued continuous function on (X, T F ) is Runiformly continuous on (X, F,τ).
9 Uniform convergence and uniform continuity Every real valued continuous function on (X, T F ) is uniformly continuous on (X, U F ). 4. (X, F,τ) is an equinormal. 5. U F is an equinormal uniformity on X. 6. The uniformity U F has the Lebesgue property. 7. (X, F,τ) has the Lebesgue property. Consider a generalized metric space (X, F,τ) with the range set (G, <). For any x X and any subset D of X we denote by F(x, D) the distance from x to D, that is, d(x, D) = inf{f(x, d) :d D}. ByF(x) we denote the distance from x to X x. Recall that a point x X is called an accumulation point of a subset S of X if F(x, S x) = 0. The accumulation points of X is denoted by A, is the set {x X : F(x) =0}. The set of isolated points is the set {x X : F(x) > 0}. Theorem 24 Let (X, F,τ) be a generalized metric spaces. statement are equivalent: The following 1. Every continuous function f : X R is uniformly continuous. 2. Every sequence {x n } in X with lim n F(x n ) = 0 has a convergent subsequence. 3. The set A is compact, and for every r 1 G there exists r 2 G such that for all x X with F(x, A) >r 1 we have F(x) >r 2. Proof. The proof is similar to that one of Theorem 1 [4]. Theorem 25 Let (X, F,τ) be a generalized metric spaces with (G, <) as a range set. Then X is compact if and only if every continuous function f : X R is uniformly continuous, and for every 0 <r G the set {x X : F(x) >r} is finite. Proof. Suppose that X is a compact space. Since on a compact space every continuous function is uniformly continuous by Theorem 10, the first condition is necessary to provide compactness of X. If for some 0 <r G the set {x X : F(x) >r} were infinite, then the family {B(x, r) ={y X : F(x, y) <r}} x X would be an open cover of X that has no finite subcover. This shows that also the second condition is necessary for compactness of X.
10 294 A. Mohamad Now assume both conditions are fulfilled, and let {x n } be any sequence in X. We have to show that {x n } has a convergent subsequence. This is trivial if some point of X occurs infinitely often in the sequence. So we may assume that no point occurs infinitely often in our sequence. Since for every 0 <r G the set {x X : F(x) >r} is finite, this implies lim n F(x n ) = 0. Thus, by Theorem 24, {x n } has a convergent subsequence. Hence, by Theorem 31, X is a compact space 5 BolzanoWeierstrass Property Definition 26 Let (X, F,τ) be a generalized metric spaces. We say that a set A X has BolzanoWeierstrass property if for every sequence {x n } of elements of A has a subsequence that converges to some x A. Definition 27 Let (X, F,τ) be a generalized metric spaces. Then X is called totally bounded if given any 0 <r G there exists a finite collection of open balls B(x 1,r),B(x 2,r),..., B(x n,r) of radius r which covers X. Lemma 28 Let (X, F,τ) be a generalized metric spaces and let K X has BolzanoWeierstrass property. If G = {G λ } is an open cover of K, then there is an 0 <r G such that for every B(k, r),k K, there exits λ Λ such that B(k, r) G λ. Proof. Suppose for every 0 <r G there is an k r K such that B(k r,r)is not subset of any G λ,λ Λ. Thus, for every n N, there is x n K such that B(x n, 1 ) is not contained in any G n λ. Now, by BolzanoWeierstrass property, there exists a subsequence {x λn } that converges to an x 0 K. Since G = {G λ } is a cover of K, there is λ 0 Λ such that x 0 G λ0. But G λ0 is open, so there exists 0 <s G such that B(x 0,s) G λ0 G λ0. By continuity of τ, there is r N such that if x B(x λr, 1 λ r ) then x B(x 0,s). Hence B(x λr, 1 λ r ) G λ, which is lead to a contradiction. Lemma 29 Let (X, F,τ) be a generalized metric spaces. X is totally bounded iff every sequence has a Cauchy subsequence. Proof. The proof is similar to that one in the classic metric spaces. Lemma 30 Let (X, F,τ) be a generalized metric spaces. Then every Cauchy sequence that clusters to a point is converges to that point.
11 Uniform convergence and uniform continuity 295 Proof. The proof is similar to that one in the classic metric spaces. Theorem 31 Let (X, F,τ) be a generalized metric spaces. statement are equivalent: The following 1. X is compact 2. X is complete and totally bounded. 3. X has the BolzanoWeierstrass property. Proof. (1) (2). Let X be a compact space. Every Cauchy sequence in X has a cluster point. So by Lemma 30, the sequence converges to that point. Hence X is complete. Also, for every r G, the open cover {B(x, r) :x X} of X, has a finite subcover. Therefore X is totally bounded. (2) (3). Since X is totally bounded, by Lemma 29, every sequence has a Cauchy subsequence. This subsequence is convergent as X is complete. Therefore X has the BolzanoWeierstrass property. (3) (1). We claim that if X has the BolzanoWeierstrass property then for every r G there exists a finite subset {x 1,x 2,..., x n } of X such that {B(x 1,r),B(x 2,r),..., B(x n,r)} covers X. Suppose not, let x 1 X, then X is not subset of B(x 1,r). Let x 2 X B(x 1,r), then {B(x 1,r),B(x 2,r)} is not covering of X. This argument leads to constrict a sequence {x n } of X such that x n+1 X n i=1 B(x i,r) for all n N. Therefore {x n } has no convergent subsequence which construct with BolzanoWeierstrass property. Hence X is compact. Corollary 32 BolzanoWeierstrass Theorem Every bounded sequence in R n has a convergent subsequence. References [1] A. George and P. Veeramani, Some Theorems in Fuzzy Metric Spaces, J. Fuzzy Math. 3(1995), [2] V. Gregori and S. Romaguera, fuzzy quasimetric spaces, Applied Gen Top 5(2004),
12 296 A. Mohamad [3] V. Gregori and S. Romaguera, Unifrom Continuity in Fuzzy Metric Spaces, Rend. Istit. Mat. Univ. Trieste 2(2001), [4] H. Hueber, On uniform continuity and compactness in metric spaces, Amer. Math. Monthly 88(1981), [5] P. S. Marcus, Probabilistic Chains, A Equations Math 15 (1977), [6] K. Menger, Statistical metrices, Proc. Nat. Acad. Sci., USA, 28 (1942), [7] A. M. Mohamad, Generalized QuasiMetric Spaces, submitted [8] A. M. Mohamad, Properties of Fuzzy QuasiMetric Spaces, Int. J. Math. Math. Sci. 2010, Art. ID , 9 pp. [9] A. M. Mohamad, Topological Study of Generalized Metric Spaces, to appear. [10] D. Kent and Richardson, Ordered Probabilistic Metric Spaces, J. Australa. Math. Soc. (series A) 46 (1989), [11] B. Schweizer and A. Sklar, Probabilistic metric spaces, NorthHolland, new York, [12] B. Schweizer, A. Sklar and E. Thorp, The metrization of SMspaces, Pacific J. Math. 10 (1960), Received: September, 2010
On Uniform Limit Theorem and Completion of Probabilistic Metric Space
Int. Journal of Math. Analysis, Vol. 8, 2014, no. 10, 455461 HIKARI Ltd, www.mhikari.com http://dx.doi.org/10.12988/ijma.2014.4120 On Uniform Limit Theorem and Completion of Probabilistic Metric Space
More informationProblem Set 2: Solutions Math 201A: Fall 2016
Problem Set 2: s Math 201A: Fall 2016 Problem 1. (a) Prove that a closed subset of a complete metric space is complete. (b) Prove that a closed subset of a compact metric space is compact. (c) Prove that
More informationChapter 2 Metric Spaces
Chapter 2 Metric Spaces The purpose of this chapter is to present a summary of some basic properties of metric and topological spaces that play an important role in the main body of the book. 2.1 Metrics
More informationPrinciples of Real Analysis I Fall VII. Sequences of Functions
21355 Principles of Real Analysis I Fall 2004 VII. Sequences of Functions In Section II, we studied sequences of real numbers. It is very useful to consider extensions of this concept. More generally,
More informationANALYSIS WORKSHEET II: METRIC SPACES
ANALYSIS WORKSHEET II: METRIC SPACES Definition 1. A metric space (X, d) is a space X of objects (called points), together with a distance function or metric d : X X [0, ), which associates to each pair
More informationAnalysis III Theorems, Propositions & Lemmas... Oh My!
Analysis III Theorems, Propositions & Lemmas... Oh My! Rob Gibson October 25, 2010 Proposition 1. If x = (x 1, x 2,...), y = (y 1, y 2,...), then is a distance. ( d(x, y) = x k y k p Proposition 2. In
More information1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3
Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,
More information7 Complete metric spaces and function spaces
7 Complete metric spaces and function spaces 7.1 Completeness Let (X, d) be a metric space. Definition 7.1. A sequence (x n ) n N in X is a Cauchy sequence if for any ɛ > 0, there is N N such that n, m
More informationSome topological properties of fuzzy cone metric spaces
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 2016, 799 805 Research Article Some topological properties of fuzzy cone metric spaces Tarkan Öner Department of Mathematics, Faculty of Sciences,
More informationREVIEW OF ESSENTIAL MATH 346 TOPICS
REVIEW OF ESSENTIAL MATH 346 TOPICS 1. AXIOMATIC STRUCTURE OF R Doğan Çömez The real number system is a complete ordered field, i.e., it is a set R which is endowed with addition and multiplication operations
More informationWeek 5 Lectures 1315
Week 5 Lectures 1315 Lecture 13 Definition 29 Let Y be a subset X. A subset A Y is open in Y if there exists an open set U in X such that A = U Y. It is not difficult to show that the collection of all
More informationAnalysis Finite and Infinite Sets The Real Numbers The Cantor Set
Analysis Finite and Infinite Sets Definition. An initial segment is {n N n n 0 }. Definition. A finite set can be put into onetoone correspondence with an initial segment. The empty set is also considered
More informationContinuous Functions on Metric Spaces
Continuous Functions on Metric Spaces Math 201A, Fall 2016 1 Continuous functions Definition 1. Let (X, d X ) and (Y, d Y ) be metric spaces. A function f : X Y is continuous at a X if for every ɛ > 0
More informationINTRODUCTION TO TOPOLOGY, MATH 141, PRACTICE PROBLEMS
INTRODUCTION TO TOPOLOGY, MATH 141, PRACTICE PROBLEMS Problem 1. Give an example of a nonmetrizable topological space. Explain. Problem 2. Introduce a topology on N by declaring that open sets are, N,
More informationMAT 544 Problem Set 2 Solutions
MAT 544 Problem Set 2 Solutions Problems. Problem 1 A metric space is separable if it contains a dense subset which is finite or countably infinite. Prove that every totally bounded metric space X is separable.
More informationg 2 (x) (1/3)M 1 = (1/3)(2/3)M.
COMPACTNESS If C R n is closed and bounded, then by BW it is sequentially compact: any sequence of points in C has a subsequence converging to a point in C Conversely, any sequentially compact C R n is
More informationPart III. 10 Topological Space Basics. Topological Spaces
Part III 10 Topological Space Basics Topological Spaces Using the metric space results above as motivation we will axiomatize the notion of being an open set to more general settings. Definition 10.1.
More informationThe ArzelàAscoli Theorem
John Nachbar Washington University March 27, 2016 The ArzelàAscoli Theorem The ArzelàAscoli Theorem gives sufficient conditions for compactness in certain function spaces. Among other things, it helps
More information4 Countability axioms
4 COUNTABILITY AXIOMS 4 Countability axioms Definition 4.1. Let X be a topological space X is said to be first countable if for any x X, there is a countable basis for the neighborhoods of x. X is said
More informationLecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University
Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University February 7, 2007 2 Contents 1 Metric Spaces 1 1.1 Basic definitions...........................
More informationFunctional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability...
Functional Analysis Franck Sueur 20182019 Contents 1 Metric spaces 1 1.1 Definitions........................................ 1 1.2 Completeness...................................... 3 1.3 Compactness......................................
More information(convex combination!). Use convexity of f and multiply by the common denominator to get. Interchanging the role of x and y, we obtain that f is ( 2M ε
1. Continuity of convex functions in normed spaces In this chapter, we consider continuity properties of realvalued convex functions defined on open convex sets in normed spaces. Recall that every infinitedimensional
More informationProblem Set 5. 2 n k. Then a nk (x) = 1+( 1)k
Problem Set 5 1. (Folland 2.43) For x [, 1), let 1 a n (x)2 n (a n (x) = or 1) be the base2 expansion of x. (If x is a dyadic rational, choose the expansion such that a n (x) = for large n.) Then the
More informationProblem Set 5: Solutions Math 201A: Fall 2016
Problem Set 5: s Math 21A: Fall 216 Problem 1. Define f : [1, ) [1, ) by f(x) = x + 1/x. Show that f(x) f(y) < x y for all x, y [1, ) with x y, but f has no fixed point. Why doesn t this example contradict
More informationMetric Spaces and Topology
Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies
More informationconverges as well if x < 1. 1 x n x n 1 1 = 2 a nx n
Solve the following 6 problems. 1. Prove that if series n=1 a nx n converges for all x such that x < 1, then the series n=1 a n xn 1 x converges as well if x < 1. n For x < 1, x n 0 as n, so there exists
More informationAssignment10. (Due 11/21) Solution: Any continuous function on a compact set is uniformly continuous.
Assignment1 (Due 11/21) 1. Consider the sequence of functions f n (x) = x n on [, 1]. (a) Show that each function f n is uniformly continuous on [, 1]. Solution: Any continuous function on a compact set
More informationFrom now on, we will represent a metric space with (X, d). Here are some examples: i=1 (x i y i ) p ) 1 p, p 1.
Chapter 1 Metric spaces 1.1 Metric and convergence We will begin with some basic concepts. Definition 1.1. (Metric space) Metric space is a set X, with a metric satisfying: 1. d(x, y) 0, d(x, y) = 0 x
More informationReal Analysis Notes. Thomas Goller
Real Analysis Notes Thomas Goller September 4, 2011 Contents 1 Abstract Measure Spaces 2 1.1 Basic Definitions........................... 2 1.2 Measurable Functions........................ 2 1.3 Integration..............................
More informationMath General Topology Fall 2012 Homework 1 Solutions
Math 535  General Topology Fall 2012 Homework 1 Solutions Definition. Let V be a (real or complex) vector space. A norm on V is a function : V R satisfying: 1. Positivity: x 0 for all x V and moreover
More informationContinuity of convex functions in normed spaces
Continuity of convex functions in normed spaces In this chapter, we consider continuity properties of realvalued convex functions defined on open convex sets in normed spaces. Recall that every infinitedimensional
More informationA continuous operator extending fuzzy ultrametrics
A continuous operator extending fuzzy ultrametrics I. Stasyuk, E.D. Tymchatyn November 30, 200 Abstract We consider the problem of simultaneous extension of fuzzy ultrametrics defined on closed subsets
More informationIntuitionistic Fuzzy Metric Groups
454 International Journal of Fuzzy Systems, Vol. 14, No. 3, September 2012 Intuitionistic Fuzzy Metric Groups Banu Pazar Varol and Halis Aygün Abstract 1 The aim of this paper is to introduce the structure
More informationReal Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi
Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.
More informationSome Basic Properties of D fuzzy metric spaces and Cantor s Intersection Theorem
Advances in Fuzzy Mathematics (AFM). ISSN 0973533X Volume 13, Number 1 (2018), pp. 49 58 Research India Publications http://www.ripublication.com/afm.htm Some Basic Properties of D fuzzy metric spaces
More informationThe HeineBorel and ArzelaAscoli Theorems
The HeineBorel and ArzelaAscoli Theorems David Jekel October 29, 2016 This paper explains two important results about compactness, the Heine Borel theorem and the ArzelaAscoli theorem. We prove them
More informationLogical Connectives and Quantifiers
Chapter 1 Logical Connectives and Quantifiers 1.1 Logical Connectives 1.2 Quantifiers 1.3 Techniques of Proof: I 1.4 Techniques of Proof: II Theorem 1. Let f be a continuous function. If 1 f(x)dx 0, then
More informationContinuity. Matt Rosenzweig
Continuity Matt Rosenzweig Contents 1 Continuity 1 1.1 Rudin Chapter 4 Exercises........................................ 1 1.1.1 Exercise 1............................................. 1 1.1.2 Exercise
More informationCourse 212: Academic Year Section 1: Metric Spaces
Course 212: Academic Year 19912 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........
More informationQUASICONTINUOUS FUNCTIONS, DENSELY CONTINUOUS FORMS AND COMPACTNESS. 1. Introduction
t m Mathematical Publications DOI: 10.1515/tmmp20170008 Tatra Mt. Math. Publ. 68 2017, 93 102 QUASICONTINUOUS FUNCTIONS, DENSELY CONTINUOUS FORMS AND COMPACTNESS L ubica Holá Dušan Holý ABSTRACT. Let
More informationTopology. Xiaolong Han. Department of Mathematics, California State University, Northridge, CA 91330, USA address:
Topology Xiaolong Han Department of Mathematics, California State University, Northridge, CA 91330, USA Email address: Xiaolong.Han@csun.edu Remark. You are entitled to a reward of 1 point toward a homework
More informationFunctional Analysis, Math 7320 Lecture Notes from August taken by Yaofeng Su
Functional Analysis, Math 7320 Lecture Notes from August 30 2016 taken by Yaofeng Su 1 Essentials of Topology 1.1 Continuity Next we recall a stronger notion of continuity: 1.1.1 Definition. Let (X, d
More informationMath 117: Topology of the Real Numbers
Math 117: Topology of the Real Numbers John Douglas Moore November 10, 2008 The goal of these notes is to highlight the most important topics presented in Chapter 3 of the text [1] and to provide a few
More informationMA651 Topology. Lecture 10. Metric Spaces.
MA65 Topology. Lecture 0. Metric Spaces. This text is based on the following books: Topology by James Dugundgji Fundamental concepts of topology by Peter O Neil Linear Algebra and Analysis by Marc Zamansky
More informationMATH 202B  Problem Set 5
MATH 202B  Problem Set 5 Walid Krichene (23265217) March 6, 2013 (5.1) Show that there exists a continuous function F : [0, 1] R which is monotonic on no interval of positive length. proof We know there
More informationMH 7500 THEOREMS. (iii) A = A; (iv) A B = A B. Theorem 5. If {A α : α Λ} is any collection of subsets of a space X, then
MH 7500 THEOREMS Definition. A topological space is an ordered pair (X, T ), where X is a set and T is a collection of subsets of X such that (i) T and X T ; (ii) U V T whenever U, V T ; (iii) U T whenever
More informationMATH 4200 HW: PROBLEM SET FOUR: METRIC SPACES
MATH 4200 HW: PROBLEM SET FOUR: METRIC SPACES PETE L. CLARK 4. Metric Spaces (no more lulz) Directions: This week, please solve any seven problems. Next week, please solve seven more. Starred parts of
More informationImmerse Metric Space Homework
Immerse Metric Space Homework (Exercises 2). In R n, define d(x, y) = x y +... + x n y n. Show that d is a metric that induces the usual topology. Sketch the basis elements when n = 2. Solution: Steps
More informationSolutions to Tutorial 8 (Week 9)
The University of Sydney School of Mathematics and Statistics Solutions to Tutorial 8 (Week 9) MATH3961: Metric Spaces (Advanced) Semester 1, 2018 Web Page: http://www.maths.usyd.edu.au/u/ug/sm/math3961/
More informationMAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9
MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended
More informationSets, Functions and Metric Spaces
Chapter 14 Sets, Functions and Metric Spaces 14.1 Functions and sets 14.1.1 The function concept Definition 14.1 Let us consider two sets A and B whose elements may be any objects whatsoever. Suppose that
More informationMAS3706 Topology. Revision Lectures, May I do not answer enquiries as to what material will be in the exam.
MAS3706 Topology Revision Lectures, May 208 Z.A.Lykova It is essential that you read and try to understand the lecture notes from the beginning to the end. Many questions from the exam paper will be similar
More informationMid Term1 : Practice problems
Mid Term1 : Practice problems These problems are meant only to provide practice; they do not necessarily reflect the difficulty level of the problems in the exam. The actual exam problems are likely to
More informationBootcamp. Christoph Thiele. Summer As in the case of separability we have the following two observations: Lemma 1 Finite sets are compact.
Bootcamp Christoph Thiele Summer 212.1 Compactness Definition 1 A metric space is called compact, if every cover of the space has a finite subcover. As in the case of separability we have the following
More informationSet, functions and Euclidean space. Seungjin Han
Set, functions and Euclidean space Seungjin Han September, 2018 1 Some Basics LOGIC A is necessary for B : If B holds, then A holds. B A A B is the contraposition of B A. A is sufficient for B: If A holds,
More informationFilters in Analysis and Topology
Filters in Analysis and Topology David MacIver July 1, 2004 Abstract The study of filters is a very natural way to talk about convergence in an arbitrary topological space, and carries over nicely into
More informationMcGill University Math 354: Honors Analysis 3
Practice problems McGill University Math 354: Honors Analysis 3 not for credit Problem 1. Determine whether the family of F = {f n } functions f n (x) = x n is uniformly equicontinuous. 1st Solution: The
More information16 1 Basic Facts from Functional Analysis and Banach Lattices
16 1 Basic Facts from Functional Analysis and Banach Lattices 1.2.3 Banach Steinhaus Theorem Another fundamental theorem of functional analysis is the Banach Steinhaus theorem, or the Uniform Boundedness
More informationMath 209B Homework 2
Math 29B Homework 2 Edward Burkard Note: All vector spaces are over the field F = R or C 4.6. Two Compactness Theorems. 4. Point Set Topology Exercise 6 The product of countably many sequentally compact
More informationFixed Point Theorems in Strong Fuzzy Metric Spaces Using Control Function
Volume 118 No. 6 2018, 389397 ISSN: 13118080 (printed version); ISSN: 13143395 (online version) url: http://www.ijpam.eu ijpam.eu Fixed Point Theorems in Strong Fuzzy Metric Spaces Using Control Function
More informationCOMMON FIXED POINT THEOREM IN PROBABILISTIC METRIC SPACE
Kragujevac Journal of Mathematics Volume 35 Number 3 (2011), Pages 463 470. COMMON FIXED POINT THEOREM IN PROBABILISTIC METRIC SPACE B. D. PANT, SUNNY CHAUHAN AND QAMAR ALAM Abstract. The notion of weakly
More informationPOINTWISE PRODUCTS OF UNIFORMLY CONTINUOUS FUNCTIONS
SARAJEVO JOURNAL OF MATHEMATICS Vol.1 (13) (2005), 117 127 POINTWISE PRODUCTS OF UNIFORMLY CONTINUOUS FUNCTIONS SAM B. NADLER, JR. Abstract. The problem of characterizing the metric spaces on which the
More informationMeasure Theory on Topological Spaces. Course: Prof. Tony Dorlas 2010 Typset: Cathal Ormond
Measure Theory on Topological Spaces Course: Prof. Tony Dorlas 2010 Typset: Cathal Ormond May 22, 2011 Contents 1 Introduction 2 1.1 The Riemann Integral........................................ 2 1.2 Measurable..............................................
More informationLacunary Statistical Convergence on Probabilistic Normed Spaces
Int. J. Open Problems Compt. Math., Vol. 2, No.2, June 2009 Lacunary Statistical Convergence on Probabilistic Normed Spaces Mohamad Rafi Segi Rahmat School of Applied Mathematics, The University of Nottingham
More informationSOME QUESTIONS FOR MATH 766, SPRING Question 1. Let C([0, 1]) be the set of all continuous functions on [0, 1] endowed with the norm
SOME QUESTIONS FOR MATH 766, SPRING 2016 SHUANGLIN SHAO Question 1. Let C([0, 1]) be the set of all continuous functions on [0, 1] endowed with the norm f C = sup f(x). 0 x 1 Prove that C([0, 1]) is a
More informationReal Analysis Chapter 4 Solutions Jonathan Conder
2. Let x, y X and suppose that x y. Then {x} c is open in the cofinite topology and contains y but not x. The cofinite topology on X is therefore T 1. Since X is infinite it contains two distinct points
More informationSummer JumpStart Program for Analysis, 2012 SongYing Li. 1 Lecture 7: Equicontinuity and Series of functions
Summer JumpStart Program for Analysis, 0 SongYing Li Lecture 7: Equicontinuity and Series of functions. Equicontinuity Definition. Let (X, d) be a metric space, K X and K is a compact subset of X. C(K)
More informationSOLUTIONS TO SOME PROBLEMS
23 FUNCTIONAL ANALYSIS Spring 23 SOLUTIONS TO SOME PROBLEMS Warning:These solutions may contain errors!! PREPARED BY SULEYMAN ULUSOY PROBLEM 1. Prove that a necessary and sufficient condition that the
More informationTopology, Math 581, Fall 2017 last updated: November 24, Topology 1, Math 581, Fall 2017: Notes and homework Krzysztof Chris Ciesielski
Topology, Math 581, Fall 2017 last updated: November 24, 2017 1 Topology 1, Math 581, Fall 2017: Notes and homework Krzysztof Chris Ciesielski Class of August 17: Course and syllabus overview. Topology
More informationSpaces of continuous functions
Chapter 2 Spaces of continuous functions 2.8 Baire s Category Theorem Recall that a subset A of a metric space (X, d) is dense if for all x X there is a sequence from A converging to x. An equivalent definition
More informationGeneral Topology. Summer Term Michael Kunzinger
General Topology Summer Term 2016 Michael Kunzinger michael.kunzinger@univie.ac.at Universität Wien Fakultät für Mathematik OskarMorgensternPlatz 1 A1090 Wien Preface These are lecture notes for a
More information11691 Review Guideline Real Analysis. Real Analysis.  According to Principles of Mathematical Analysis by Walter Rudin (Chapter 14)
Real Analysis  According to Principles of Mathematical Analysis by Walter Rudin (Chapter 14) 1 The Real and Complex Number Set: a collection of objects. Proper subset: if A B, then call A a proper subset
More information1. Let A R be a nonempty set that is bounded from above, and let a be the least upper bound of A. Show that there exists a sequence {a n } n N
Applied Analysis prelim July 15, 216, with solutions Solve 4 of the problems 15 and 2 of the problems 68. We will only grade the first 4 problems attempted from15 and the first 2 attempted from problems
More informationMath 328 Course Notes
Math 328 Course Notes Ian Robertson March 3, 2006 3 Properties of C[0, 1]: Supnorm and Completeness In this chapter we are going to examine the vector space of all continuous functions defined on the
More informationNotes on nets and convergence in topology
Topology I HumboldtUniversität zu Berlin C. Wendl / F. Schmäschke Summer Semester 2017 Notes on nets and convergence in topology Nets generalize the notion of sequences so that certain familiar results
More informationSome Background Material
Chapter 1 Some Background Material In the first chapter, we present a quick review of elementary  but important  material as a way of dipping our toes in the water. This chapter also introduces important
More informationExam 2 extra practice problems
Exam 2 extra practice problems (1) If (X, d) is connected and f : X R is a continuous function such that f(x) = 1 for all x X, show that f must be constant. Solution: Since f(x) = 1 for every x X, either
More information2. Metric Spaces. 2.1 Definitions etc.
2. Metric Spaces 2.1 Definitions etc. The procedure in Section for regarding R as a topological space may be generalized to many other sets in which there is some kind of distance (formally, sets with
More information4. (alternate topological criterion) For each closed set V Y, its preimage f 1 (V ) is closed in X.
Chapter 2 Functions 2.1 Continuous functions Definition 2.1. Let (X, d X ) and (Y,d Y ) be metric spaces. A function f : X! Y is continuous at x 2 X if for each " > 0 there exists >0 with the property
More informationTHEOREMS, ETC., FOR MATH 515
THEOREMS, ETC., FOR MATH 515 Proposition 1 (=comment on page 17). If A is an algebra, then any finite union or finite intersection of sets in A is also in A. Proposition 2 (=Proposition 1.1). For every
More informationEconomics 204 Fall 2011 Problem Set 2 Suggested Solutions
Economics 24 Fall 211 Problem Set 2 Suggested Solutions 1. Determine whether the following sets are open, closed, both or neither under the topology induced by the usual metric. (Hint: think about limit
More informationA PROOF OF A CONVEXVALUED SELECTION THEOREM WITH THE CODOMAIN OF A FRÉCHET SPACE. MyungHyun Cho and JunHui Kim. 1. Introduction
Comm. Korean Math. Soc. 16 (2001), No. 2, pp. 277 285 A PROOF OF A CONVEXVALUED SELECTION THEOREM WITH THE CODOMAIN OF A FRÉCHET SPACE MyungHyun Cho and JunHui Kim Abstract. The purpose of this paper
More informationKirk s Fixed Point Theorem in Generating Spaces of SemiNorm Family
Gen. Math. Notes, Vol. 21, No. 2, April 2014, pp.113 ISSN 22197184; Copyright c ICSRS Publication, 2014 www.icsrs.org Available free online at http://www.geman.in Kirk s Fixed Point Theorem in Generating
More informationMATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5
MATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5.. The ArzelaAscoli Theorem.. The Riemann mapping theorem Let X be a metric space, and let F be a family of continuous complexvalued functions on X. We have
More informationAnalysis Comprehensive Exam Questions Fall F(x) = 1 x. f(t)dt. t 1 2. tf 2 (t)dt. and g(t, x) = 2 t. 2 t
Analysis Comprehensive Exam Questions Fall 2. Let f L 2 (, ) be given. (a) Prove that ( x 2 f(t) dt) 2 x x t f(t) 2 dt. (b) Given part (a), prove that F L 2 (, ) 2 f L 2 (, ), where F(x) = x (a) Using
More informationMath 4317 : Real Analysis I MidTerm Exam 1 25 September 2012
Instructions: Answer all of the problems. Math 4317 : Real Analysis I MidTerm Exam 1 25 September 2012 Definitions (2 points each) 1. State the definition of a metric space. A metric space (X, d) is set
More informationTools from Lebesgue integration
Tools from Lebesgue integration E.P. van den Ban Fall 2005 Introduction In these notes we describe some of the basic tools from the theory of Lebesgue integration. Definitions and results will be given
More informationAnalysis3 lecture schemes
Analysis3 lecture schemes (with Homeworks) 1 Csörgő István November, 2015 1 A jegyzet az ELTE Informatikai Kar 2015. évi Jegyzetpályázatának támogatásával készült Contents 1. Lesson 1 4 1.1. The Space
More informationCommon Fixed Point Theorem Satisfying Implicit Relation On Menger Space.
www.ijecs.in International Journal Of Engineering And Computer Science ISSN:23197242 Volume 5 Issue 09 September 2016 Page No.1818018185 Common Fixed Point Theorem Satisfying Implicit Relation On Menger
More informationMath 5210, Definitions and Theorems on Metric Spaces
Math 5210, Definitions and Theorems on Metric Spaces Let (X, d) be a metric space. We will use the following definitions (see Rudin, chap 2, particularly 2.18) 1. Let p X and r R, r > 0, The ball of radius
More informationMetric Spaces. DEF. If (X; d) is a metric space and E is a nonempty subset, then (E; d) is also a metric space, called a subspace of X:
Metric Spaces DEF. A metric space X or (X; d) is a nonempty set X together with a function d : X X! [0; 1) such that for all x; y; and z in X : 1. d (x; y) 0 with equality i x = y 2. d (x; y) = d (y; x)
More information2 Metric Spaces Definitions Exotic Examples... 3
Contents 1 Vector Spaces and Norms 1 2 Metric Spaces 2 2.1 Definitions.......................................... 2 2.2 Exotic Examples...................................... 3 3 Topologies 4 3.1 Open Sets..........................................
More informationCHAPTER 1. Metric Spaces. 1. Definition and examples
CHAPTER Metric Spaces. Definition and examples Metric spaces generalize and clarify the notion of distance in the real line. The definitions will provide us with a useful tool for more general applications
More informationA NOTE ON ΘCLOSED SETS AND INVERSE LIMITS
An. Şt. Univ. Ovidius Constanţa Vol. 18(2), 2010, 161 172 A NOTE ON ΘCLOSED SETS AND INVERSE LIMITS Ivan Lončar Abstract For every Hausdorff space X the space X Θ is introduced. If X is Hclosed, then
More informationMetric Spaces Math 413 Honors Project
Metric Spaces Math 413 Honors Project 1 Metric Spaces Definition 1.1 Let X be a set. A metric on X is a function d : X X R such that for all x, y, z X: i) d(x, y) = d(y, x); ii) d(x, y) = 0 if and only
More informationLecture 5  Hausdorff and GromovHausdorff Distance
Lecture 5  Hausdorff and GromovHausdorff Distance August 1, 2011 1 Definition and Basic Properties Given a metric space X, the set of closed sets of X supports a metric, the Hausdorff metric. If A is
More informationA Common Fixed Point Theorems in Menger(PQM) Spaces with Using Property (E.A)
Int. J. Contemp. Math. Sciences, Vol. 6, 2011, no. 4, 161167 A Common Fixed Point Theorems in Menger(PQM) Spaces with Using Property (E.A) Somayeh Ghayekhloo Member of young Researchers club, Islamic
More informationMath 118B Solutions. Charles Martin. March 6, d i (x i, y i ) + d i (y i, z i ) = d(x, y) + d(y, z). i=1
Math 8B Solutions Charles Martin March 6, Homework Problems. Let (X i, d i ), i n, be finitely many metric spaces. Construct a metric on the product space X = X X n. Proof. Denote points in X as x = (x,
More informationProduct metrics and boundedness
@ Applied General Topology c Universidad Politécnica de Valencia Volume 9, No. 1, 2008 pp. 133142 Product metrics and boundedness Gerald Beer Abstract. This paper looks at some possible ways of equipping
More informationSection 21. The Metric Topology (Continued)
21. The Metric Topology (cont.) 1 Section 21. The Metric Topology (Continued) Note. In this section we give a number of results for metric spaces which are familar from calculus and real analysis. We also
More information