Conditions (b1) and (b2) also appear in the definition of metric space. How many of the following subsets SR2S \subset \mathbb{R}^2SR2 are closed in this metric space? A metric space consists of a set M of arbitrary elements, called points, between which a distance is defined i.e. Metric spaces: basic definitions Let Xbe a set. (Open Set, Closed Set, Neighbourhood.) Thus, limnxn=x\lim_{n\to\infty} x_n = xlimnxn=x. A non-empty set Y of X is said to be compact if it is compact as a metric space. So is now a completion. Samet et al. )%d1ObkB8M \KDRmT{,PK>EyH1N# G[Dd]nlZj]FzONSHTYGIhpL{`1>|Vv;*o RqN#K['b stream In calculus, there is a notion of convergence of sequences: a sequence {xn}\{x_n\}{xn} converges to xxx if xnx_nxn gets very close to xxx as nnn approaches infinity. Denote and as the sets of all real and natural numbers, respectively. In Rn\mathbb{R}^nRn, the Euclidean distance between two points x=(x1,,xn)\mathbf{x} = (x_1, \cdots, x_n)x=(x1,,xn) and y=(y1,,yn)\mathbf{y} = (y_1, \cdots, y_n)y=(y1,,yn) is defined to be xy=i=1n(xiyi)2.\| \mathbf{x} - \mathbf{y} \| = \sqrt { \sum_{i=1}^{n} (x_i - y_i)^2 }. The metric \(d\) is called the discrete metricand \((M,d)\) is a discrete space. Examples include distributions in L^2-Wasserstein space . Also defined as Some sources place no emphasis on the fact that the subset B of the underlying set A of M is in fact itself a subspace of M , and merely refer to a bounded set . >> endobj These can be downloaded below. In our main theorems, we employ a binary relation on the metric space, which does not have to be a partial order. Some important properties of this idea are abstracted into: d ( x, y) + d ( y, z) d ( x, z ). endobj Define a sequence by . Answer (1 of 3): Topological space is the generalized form of metric space. Lemma 2 For help downloading and using course materials, read our FAQs . This paper is structured as follows: In 2, we show a brief review of 4D EGB gravity. A metric space $ (X,\rho)$ is compact if and only if it is complete and totally bounded, and $ (X,\rho)$ is totally bounded if and only if it is isometric to a subset of some compact metric space. The set SSS is called closed if S=SS = \overline{S}S=S. >> Example 4. While every effort has been made to follow citation style rules, there may be some discrepancies. A contraction is a function f:MMf: M \to Mf:MM for which there exists some constant 0 Let us know if you have suggestions to improve this article (requires login). 48 0 obj << Consider a closed subset CYC \subset YCY, so that CCC contains all points near it. /Subtype /Link A sequence {xn}M\{x_n\} \subset M{xn}M is called Cauchy if, for every >0\epsilon > 0>0, there exists an index NNN_{\epsilon} \in \mathbb{N}NN such that whenever m,nNm, n \ge N_{\epsilon}m,nN, the inequality d(xm,xn)> endobj If {xn}\{x_n\}{xn} has a convergent subsequence, then {xn}\{x_n\}{xn} converges. Let Y be a nonempty subset of X in a metric space (X, p). The distance from a to b is | a - b |. This space (X;d) is called a discrete metric space. Let M = (X, d) be a metric space . Within this manuscript we generalize the two recently obtained results of O. Popescu and G. Stan, regarding the F-contractions in complete, ordinary metric space to 0-complete partial metric space and 0-complete metric-like space. 1. /Type /Annot De nition 1.2. By the triangle inequality, d(x,x)d(x,y)+d(y,x)d(x,x) \le d(x,y) + d(y,x)d(x,x)d(x,y)+d(y,x). Required fields are marked *, \(\begin{array}{l}E\subseteq \bar{E}\end{array} \), \(\begin{array}{l}E= \bar{E}\end{array} \), \(\begin{array}{l}\bar{B}(x, r)\equiv \left\{x\in X | p(x, x)\leq r \right\}\end{array} \), \(\begin{array}{l}\bar{B}(x, r)\end{array} \), \(\begin{array}{l}\bar{B}(0, 1)\end{array} \), \(\begin{array}{l}\displaystyle \lim_{ n\to \infty 0}p(x_{n}, x)=0\end{array} \), \(\begin{array}{l}d'(x, y)= \frac{d(x, y)}{1+d(x, y)}, (x, y\in X)\end{array} \), \(\begin{array}{l}1+b+c \leq (1+b)(1+c)\end{array} \), \(\begin{array}{l}\frac{2+b+c}{(1+b)(1+c)}\leq \frac{2+b+c}{1+b+c}\end{array} \), \(\begin{array}{l}\frac{1}{1+b}+\frac{1}{1+c}\leq 1+\frac{1}{1+b+c}\leq 1+\frac{1}{1+a}\end{array} \), \(\begin{array}{l}1-\frac{1}{1+a}\leq \left ( 1-\frac{1}{1+b} \right )+\left ( 1-\frac{1}{1+c} \right )\end{array} \), \(\begin{array}{l}\frac{a}{1+a}\leq \frac{b}{1+b}+\frac{c}{1+c}\end{array} \), \(\begin{array}{l}d'(x, y)= \frac{d(x, y)}{1+d(x, y)}\end{array} \), Frequently Asked Questions on Metric Spaces. Suppose satisfies the first two conditions. We strive to present a forum where all aspects of these problems can be discussed. Knowing whether or not a metric space is complete is very useful, and many common metric spaces are complete. >> endobj 36 0 obj << In addition, some applications of the main results to continuous data dependence of the fixed points of operators defined on these spaces were shown. If f:MMf: M \to Mf:MM is a contraction, does fff have a fixed point? <> No Resources Found. Exercise 1. Property 2 states if the distance between x and y equals zero, it is because we are considering the same point. Corrections? If f:MMf: M \to Mf:MM has a fixed point, is fff a contraction? conditions: A pair , where is a metric on is called a metric space. The metric space X is said to be compact if every open covering has a nite subcovering.1 This abstracts the Heine-Borel property; indeed, the Heine-Borel theorem states that closed bounded subsets of the real line are compact. $\endgroup$ - Joseph Van Name ~"K:dN) XoQV4FUs5XKJV@U*_pze}{>nG3`vSMj*pO]XEj?aOZXT=8~ #$ t| (ii) . Ecological conditions predict the intensity of Hendra virus excretion over space and time from bat reservoir hosts Published in: Ecology Letters, October 2022 DOI: 10.1111/ele.14007: Pubmed ID: 36310377. X is dense in Y. Equivalently, {xn}\{x_n\}{xn} is Cauchy if and only if limmlimnd(xm,xn)=0.\lim_{m\to\infty} \lim_{n\to\infty} d(x_m, x_n) = 0.mlimnlimd(xm,xn)=0. In general topology, there are two common types of sets, open sets and closed sets. Discrete metric space is often used as (extremely useful) counterexamples to illustrate certain concepts. Hence, we can say that d is a metric on X. d (x, y) = 0 if and only if x = y. d is called a metric, and d (x, y )is the distance from x to y. This is the usual distance used in Rm, and when we speak about Rm as a metric space without specifying a metric, it's the Euclidean metric that is intended. Working off this definition, one is able to define continuous functions in arbitrary metric spaces. Let (M,d)(M,d)(M,d) be a complete metric space. This says that d 1(x,y) is the distance from x to y if you can only travel along rays through the origin. /A << /S /GoTo /D (subsection.1.2) >> Parametric metric space is the generalization of metric space too. d: X X R +. Moreover, has a unique fixed point when or for all . The second approach is much easier and more organized, so the concept of a metric space was born. Given that X is a metric space, with the metric d. Define. Then d stream In this paper, we consider common fixed point theorems in the framework of the refined cone metric space, namely, quasi-cone metric space. A metric space is a set equipped with a distance function, which provides a measure of distance between any two points in the set. 9 0 obj 33 0 obj << The distance function, known as a metric, must satisfy a collection of axioms. (3) The pairs (R2,dE)\big(\mathbb{R}^2, d_{E}\big)(R2,dE) and (R2,dT)\big(\mathbb{R}^2, d_{T}\big)(R2,dT) are both metric spaces. /A << /S /GoTo /D (subsection.1.4) >> The usual metric on the rational numbers is not complete since some Cauchy sequences of rational numbers do not converge to rational numbers. endobj (2) f is called a Quasi-isometry, if f satisfies, for some B, b > 0 and all x, y X and in addition f ( X) is C -dense. xy=i=1n(xiyi)2. >> endobj Suppose {x n} is a convergent sequence which converges to two dierent limits x 6= y. Section 3 builds on the ideas from the first two sections to formulate a definition of continuity for functions between metric spaces. Metric Space - Revisited. endobj ?{8BxWMZ?fF7_w7oyjqjLha8j/ /\;7 3p,v Induced topology The natural generalization of continuity for real-valued functions of a real variable is as follows: At the point xX provided for any sequence {xn} in X, a mapping f from a metric space X to a metric space Y is also said to be continuous. A norm is a nonnegative real-valued function ||.|| on a linear space X for any u, v X and real number a, A normed linear space is a linear space that has a norm. The triangle inequality property for the metric is given by: p(x, y) p(x, z) + p(z, y). Triangular norms are used to generalize with the probability distribution of triangle inequality in metric space conditions. Lemma: Let (M,d)(M, d)(M,d) be a metric space. /A << /S /GoTo /D (subsection.1.6) >> Intuitively, if a function f:XYf: X \to Yf:XY is continuous, it should map points that are near one another in XXX to points that are near one another in YYY. [19] introduced the class of -admissible mappings on metric spaces and the concept of (-)-contractive mapping on complete metric spaces and established some fixed point theorems . Then has a unique coupled fixed point. It covers metrics, open and closed sets, continuous functions (in the topological sense), function spaces, completeness, and compactness. Abstract: Adams inequalities with exact growth conditions are derived for Riesz-like potentials on metric measure spaces. A neighbourhood of x for a point x X is an open set that includes x. Log in. )f(x) = x?)f(x)=x?). A sequence in a metric space is called Cauchy if for every positive real number there is a positive integer such that for all positive integers Complete space A metric space is complete if any of the following equivalent conditions are satisfied: Every Cauchy sequence of points in has a limit that is also in Every Cauchy sequence in converges in Example 1: If we let d(x,y) = |xy|, (R,d) is a metric . Which was introduced and studied by Hussian (a new approach to metric space) in 2014. This result may appear obscure and uninteresting, but the payoff is actually glorious: one can use it to prove the existence of solutions to all ordinary differential equations! A metric space is a non-empty set equi pped with structure determined by a well-defin ed notion of distan ce. Forgot password? The most familiar example of a metric space is 3-dimensional Euclidean space with . endobj %PDF-1.5 T = { A X a A: > 0: B d ( a, ) A } I.e, the topology on X is induced by a metric. Since Tis a triangular admissible mapping, then or . A natural question is when are two metric spaces. >> endobj 3 0 obj % /Type /Annot - discrete metric. Updates? (Metric Space.) The pair (X;d) is called a metric space. /Length 392 It is clearly given that d(x, y) = 0, if and only of d(x, y) = 0. xn0RU=` Ho_o) ZzD^]>S3 Bb &Z3Ph%\ A metric space is called sequentially compact if every sequence of elements of has a limit point in . This is a weaker concept than that of a metric space. A distance function satisfying all the above three conditions is termed a metric . Given any set of points, the discrete metric specifies that the distance from a point to itself equal 0 while the distance between any two distinct points equal 1. A metric space (X,d) is a set X with a metric d dened on X. In this paper, we have provided some fixed point results for self-mappings fulfilling generalized contractive conditions on altered metric spaces. << /S /GoTo /D (subsection.1.1) >> A topological space ( X, T) is called metrizable if there exists a metric. Amar Kumar Banerjee, Sukila Khatun. Indeed, the R-charges of fields may be computed usinga-maximisation [14], and agree with the . Theorem: A closed ball is a closed set. The constraint of p to Y Y thus defines a metric on Y, which we refer to as a metric subspace. Let such that or . 35 0 obj << endobj Let be a metric space and a functional. 17 0 obj /Rect [88.563 641.654 268.417 654.273] endobj The proof of this fact, given in 1914 by the German mathematician Felix Hausdorff, can be generalized to demonstrate that every metric space has such a completion. The closure S\overline{S}S of SSS is S:={yM:d(y,S)=0}.\overline{S}:= \{y \in M \, : \, d(y, S) = 0 \}.S:={yM:d(y,S)=0}. In present paper we prove two fixed point theorems based on injective mapping using contraction conditions. >> Dividing this by two gives the desired result. In general, the theorem has been generalized in two directions. FGC}| {]XxMiUov/mES) 37 0 obj << However, the usual metric on the real numbers is complete, and, moreover, every real number is the limit of a Cauchy sequence of rational numbers. Sign up, Existing user? New user? Hence, a metric space is a nonempty subset of Euclidean space, of an LP(E) space,1 p , and of C[a, b]. For example, the axioms imply that the distance between two points is never negative. https://www.britannica.com/science/metric-space. /Type /Annot dE((x1,y1),(x2,y2))=(x1x2)2+(y1y2)2. Formally, a metric space is a pair M = (X,d) where X is a finite set of size N nodes, equipped with the distance metric function d: X X R+; for each a,b X the distance between a and b is given by the function d(a,b). Assume that b, c are the non-negative numbers. In what follows, we shall recall the basic . The usual distance function on the real number line is a metric, as is the usual distance function in Euclidean n-dimensional space. We know d(x,x)=0d(x,x) = 0d(x,x)=0 and d(x,y)=d(y,x)d(x,y) = d(y,x)d(x,y)=d(y,x), so this inequality implies 2d(x,y)02d(x,y) \ge 02d(x,y)0. 42 0 obj << The following problem concerns the Banach fixed point theorem, an abstract result about complete metric spaces. xZ[~_RyrA M+qt]?wW Ui\XQ99skf\\,_ejq^eWwes_0*|[zq$BjwE1 $n&ITM2j[A [%neXia$\:%m J%<2,!%J^%|{5&gpDDv %QJK,uVvpRQ'y*D:7y.W5DP?SQIf8@_FISm9d%1b&{:_t;DNCpT,{_\h*knw&)F]kOEca Choose and set , , . Then we take Now Z is closed in Y so it is complete by proposition 1 above. >> endobj More precisely, total boundedness of a metric space is equivalent to compactness of its completion $ (\hat X,\tilde\rho)$. As expected, the basic topological notions were defined analogously. 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Log in here. fWx~ In 3 we discuss the wormhole metric and solution of field equations in 4D EGB gravity. Conditions (1) and (2) are similar to the metric space, but (3) is a key feature of this concept. We can rephrase compactness in terms of closed sets by making the following observation: (Further Examples of Metric Spaces.) For example, a finite set in any metric space (X, d) is compact. (Convergence, Cauchy Sequence, Completeness.) This taxicab distance gives the minimum length of a path from (x,y) to (z,w) constructed from horizontal and vertical line segments. In this way metric spaces provide important examples of topological spaces. In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points.The distance is measured by a function called a metric or distance function. In a metric space, we can measure nearness using the metric, so closed sets have a very intuitive definition. If the mapping f is continuous at every point in X, it is said to be continuous. A metric space is a set equipped with a distance function, which provides a measure of distance between any two points in the set. zn6'}v=WG\W67Z8ZD6/5 R[,y0Z For any convergent sequence xnxx_n \to xxnx, the points xnx_nxn and xmx_mxm are very close for large mmm and nnn, since both points are known to be close to xxx. /Font << /F18 43 0 R /F15 44 0 R >> Any subset of with the same metric. RM Theorem: A subset A of a metric space is closed if and only if its complement Ac A c is open. 21 0 obj And the probabilistic metric space is one of the important generalizations of metric space introduced by Austrian mathematician Karl Menger in 1942. The term metric space is frequently denoted (X, p). Sign up to read all wikis and quizzes in math, science, and engineering topics. The core of this package is Frchet regression for random objects with Euclidean predictors, which allows one to perform regression analysis for non-Euclidean responses under some mild conditions. Since {xn}\{x_n\}{xn} is Cauchy, we can also choose MMM such that m,nMm, n \ge Mm,nM implies d(xn,xm)<2d(x_n, x_m) < \frac{\epsilon}2d(xn,xm)<2. For instance, R\mathbb{R}R is complete under the standard absolute value metric, although this is not so easy to prove. /Contents 39 0 R Suppose (X, p) be a metric space. 28 0 obj /Parent 45 0 R Sequential compactness is equivalent to topological compactness on metric spaces. /Border[0 0 1]/H/I/C[1 0 0] metric space, in mathematics, especially topology, an abstract set with a distance function, called a metric, that specifies a nonnegative distance between any two of its points in such a way that the following properties hold: (1) the distance from the first point to the second equals zero if and only if the points are the same, (2) the distance from the first point to the second equals the distance from the second to the first, and (3) the sum of the distance from the first point to the second and the distance from the second point to a third exceeds or equals the distance from the first to the third. Section 2 develops the idea of sequences and convergence in metric spaces. A function p: X X R is known as a metric provided for all x, y, and z in X. In this sense, the real numbers form the completion of the rational numbers. Indeed, a space-based laboratory can ensure long free-fall conditions and long interaction times, important for precision tests . We note that in the case where K = 1, every b -metric space is obviously a metric space. Let M = (Y, dY) be a subspace of M . Let JJJ be an index such that kJk\ge JkJ implies nkMn_k \ge MnkM; this exists simply because {nk}\{n_k\}{nk} is a strictly increasing sequence of positive integers. But CCC contains all points near it, so f(x)Cf(x) \in Cf(x)C, and hence xf1(C)x\in f^{-1} (C)xf1(C). endobj An ultrametric space is a pair (M, d) consisting of a set M together with an ultrametric d on M, which is called the space's associated distance function (also called a metric). Lecture 4: Compact Metric Spaces. 30 0 obj << Already, one can see that these axioms imply results that are consistent with intuition about distances. Solution: For any x;y2X= R, the function d(x;y) = jx yjde nes a metric on X= R. It can be easily veri ed that the absolute value function . /Border[0 0 1]/H/I/C[1 0 0] Then every open ball B(x;r) B ( x; r) with centre x contain an infinite numbers of point of A. Space provides the ideal conditions for testing fundamental physics. As, Now, add 1 to both sides of the inequality and rearrange the terms, we get. /Filter /FlateDecode /Rect [88.563 670.546 357.257 683.165] Prove that condition 1 follows from conditions 2-4. In recent years, a number of fixed point results for single-valued and multi-valued operators in b-metric spaces have been studied extensively in [4-6, 10-12, 17, 20] and . Property 1 expresses that the distance between two points is always larger than or equal to 0. i.e.,, for each > 0, there should be an index N such that n > N, p(xn, x) < . The well-known example of metric space is the set R of all real numbers with p(x, y) = | x y |. Such that a b+c, for any non-negative numbers a, b and c. Hence, for x, y, z X, the non-negative numbers, such as a = d(x, y), b = d(x, z) and c = d(z, y) satisfies the condition a b+c for the metric d, by the triangle inequality. Show that the real line is a metric space. In the present work, we are motivated to find non-exotic wormholes in 4D EGB gravity, i.e. Your Mobile number and Email id will not be published. Let Xbe any set, and de ne the function d: X X!R by >> endobj <>>> A point x X is called a point of closure of E for a subset E of a metric space X if every neighbourhood of x includes a point in E. The closure of E is the set of Es points of closure and is represented by, In metric space (X, p), a sequence {xn} is said to converge to the point xX, given. Completeness Proofs.) Normed Spaces- a subsection of metric spaces. M=RnM = \mathbb{R}^nM=Rn and d((x1,,xn),(y1,,yn))=max1inxiyid\big((x_1, \ldots, x_n), (y_1, \ldots, y_n)\big) = \max_{1\le i \le n} |x_i - y_i|d((x1,,xn),(y1,,yn))=1inmaxxiyi, M={a,b,c,d},M = \{a, b, c, d\},M={a,b,c,d}, where d(a,b)=d(a,c)=3d(a,b) = d(a,c) = 3d(a,b)=d(a,c)=3, d(a,d)=d(b,c)=7d(a,d) = d(b,c) = 7d(a,d)=d(b,c)=7, and d(b,d)=d(c,d)=11d(b,d) = d(c,d) = 11d(b,d)=d(c,d)=11, M=C[0,1]M = \mathcal{C}[0,1]M=C[0,1], the set of continuous functions [0,1]R[0,1] \to \mathbb{R}[0,1]R, and d(f,g)=maxx[0,1]f(x)g(x)d(f,g) = \max_{x\in [0,1]} |f(x) - g(x)|d(f,g)=x[0,1]maxf(x)g(x), M=C[0,1]M = \mathcal{C}[0,1]M=C[0,1] and d(f,g)=01(f(x)g(x))2dxd(f,g) = \int_{0}^{1} \big(f(x) - g(x)\big)^2 \, dxd(f,g)=01(f(x)g(x))2dx. ConsiderX =C n, with a weightedC action with weights v(Z +) n.The orbit space of non-zero vectors is the weighted projective spaceWCP n 1 [v1,.,v n].Existence of a Ricci- flat Kahler cone metric onC n, with the conical symmetry generated by thisC action, is equivalent to existence of a Kahler-Einstein orbifold metric on the weighted projective space. /A << /S /GoTo /D (subsection.1.5) >> Comment: When it is clear or irrelevant which metric d we have in mind, we shall often refer to "the metric space X" rather than "the metric space (X,d)". A metric space is said to be complete if every sequence of points in which the terms are eventually pairwise arbitrarily close to each other (a so-called Cauchy sequence) converges to a point in the metric space. For instance, the higher dimensional Euclidean spaces Rn\mathbb{R}^nRn and the circle all have their own notions of distance. We can dene many dierent metrics on the same set, but if the metric on X is clear from the context, we refer to X as a metric space and omit explicit mention of the metric d. Example 7.2. The French mathematician Maurice Frchet initiated the study of metric spaces in 1905. Equivalently: every sequence has a converging sequence. (i) if and only if . endobj Mnqn:u6 86E1N@{0S;nm==72NOrLoUGc%;p{qgxi2|ygII[KA%9K# D6:!F*gD3RQnQHtxlc4 W pixArTZXiYDa99Ap30>A.;o;X7U9x. Intuitively, an open set is a set that does not contain its boundary; the endpoints of an interval are not contained in the interval. A rather trivial example of a metric on any set X is the discrete metric d(x,y) = {0 if x . Suppose that the mapping satisfies the following contractive condition for all : (2.1) where are nonnegative constants with . 4 0 obj endstream In [18] a family of supersymmetric quiver gauge theories were studied whose classical vacuum moduli space reproduces the affine varieties X2p.These theories were argued to flow for large N in the IR to a superconformal fixed point, AdS/CFT dual to a Sasaki-Einstein metric on the linkL2p for allp. Your Mobile number and Email id will not be published. #,P+R: P76M31^OXSbEVSK79p:|D,84zg #p*\>wIALj q9IfXS`q}g! For example, the rational number sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, converges to , which is not a rational number. In 1976, Caristi defined an order relation in a metric space by using a functional under certain conditions and proved a fixed point theorem for such an ordered metric space. 8 0 obj candiates for isomorphism. We will call such spaces semi-metric spaces. Let be a complete cone metric space. /Rect [88.563 656.1 284.87 668.719] It is straightforward to show that \((M,d)\) is a metric space. In what follows, assume (M,d)(M,d)(M,d) is a metric space. Let us take a closer look at the various concepts associated with metric spaces in this article. So this concept is a weaker concept than that of a metric space. The third condition is a consequence of the inequality jjx+yjj jjxjj+jjyjj(replace x and G+dv ,*8ZZW\2}eM`. >> endobj Then Thas a fixed point. << /S /GoTo /D (subsection.1.2) >> << /S /GoTo /D (subsection.1.3) >> 29 0 obj The pair is called an . In analysis there are several useful metrics on sets of bounded real-valued continuous or integrable functions. <>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 594.6 843.24] /Contents 4 0 R/Group<>/Tabs/S/StructParents 1>> Hint: The first question is much harder than the second. We require that for all a,b,c X, (Antireflexivity) d(a,b) = 0 if and only if a = b (Symmetry) d(a,b) = d(b,a) endobj In many applications of the journal of mathematical sciences, on the other hand, metric space has a metric derived from a norm that determines the "length" of a vector. /A << /S /GoTo /D (section.1) >> This is easily shown to be a metric; it is known as the standard discrete metric on S. (3) Let d be the Euclidean metric on R3, and for x, y R3 dene d(x,y) = d(x,y) if x = sy or y = sx for some s R d(x,0)+ d(0,y) otherwise. /Rect [88.563 684.991 315.241 697.611] 12 0 obj It is clear that an extended b -metric space coincides with the corresponding b -metric space, for \theta (x,y)=s \geq 1 where s\in \mathbb {R} and it turns to be standard metric if s=1. 16 0 obj Choose >0\epsilon > 0>0 and KKK such that kKk\ge KkK implies d(xnk,x)<2d(x_{n_k}, x) < \frac{\epsilon}2d(xnk,x)<2. Course Description This course provides a basic introduction to metric spaces. In fact, the answer is yes, and this extremely important result is known as the Banach fixed point theorem.
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