Xxxr is called a metric or distance function if ad only if. A pair, where is a metric on is called a metric space. Cauchy sequences 1 here we begin with the following. Quasiisometry of metric spaces is an equivalence relation. Quasimetrics are a subject of comprehensive investigation both in pure and applied mathematics in areas such as in functional analysis, topology and computer science. As applications, some previous results are obtained from these theorems as special cases. Let f be a monotone mapping from a smyth complete quasi metric space x, q into itself such that. We study the geometry of nonrelatively hyperbolic groups. Topology and its applications 30 1988 127148 127 northholland on completeness in quasi metric spaces doitchin doitchinov department of mathematics, university of sofia, 1090 sofia, bulgaria received 5 september 1986 revised 24 august 1987 a notion of cauchy sequence in quasi metric spaces is introduced and used to define a standard completion for a special class of spaces. Thereafter, we investigate not only the existence, but also the uniqueness of a fixed point of such mappings in the setting of rightcomplete quasimetric spaces. In particular, applications to two interesting examples of partial metric spaces which appear in computer science, as the domain of words and. This paper contains a study of families of quasi pseudometrics the concept of a quasi pseudo metric was introduced by wilson 22, albert 1 and kelly 9 generated by proba. Romaguera, quasimetric spaces, quasimetric hyperspaces and uniform local compactness, in.
Fixed point theorems for fuzzy contractive mappings in. An international journal of mathematics, 30 1999 suppl. X x is calledgeneralized contraction if and only if for all x,y. In this paper, we obtain some fixed point theorems for fuzzy mappings in a left ksequentially complete quasipseudometric space and in a right ksequentially complete quasipseudometric space, respectively. The concept of quasimetric spaces was introduced by wilson in 1931 as a generalization of metric spaces, and in 2000 hitzler and seda introduced dislocated metric spaces as a. Since such spaces are quasimetrizable but nonmetrizable, we will need to develop our theory in the realm of quasimetric spaces. Hardy spaces on ahlforsregular quasi metric spaces a.
Notes on metric spaces these notes are an alternative to the textbook, from and including closed sets and open sets page 58 to and excluding cantor sets page 95 1 the topology of metric spaces assume m is a metric space with distance function d. Ais a family of sets in cindexed by some index set a,then a o c. As for the box metric, the taxicab metric can be generalized to rnfor any n. By using a suitable modification of the notion of a distance we obtain some fixed point results for generalized contractive setvalued maps on complete preordered quasimetric spaces. The next result can be stated when one considers a quasimetric space, a particular instance of partial quasimetric spaces, in the statement of theorem 6.
Some coupled fixed point theorems on quasipartial bmetric. It is evident that any metric space is a quasimetric space, but the converse is not true in general. Chapter 9 the topology of metric spaces uci mathematics. The monograph is largely selfcontained and is intended for mathematicians, graduate students and professionals with a mathematical background who are interested in the interplay between. From this result we deduce that any partial metric space has an up to isometry unique partial metric bicompletion. They called a t 0quasimetric space qhyperconvex if and only if it is injective in the category of t 0. Again in many respects completeness and together with it. Some coupled xed point theorems on quasipartial bmetric spaces 297 4 the main results theorem 4. Note that iff if then so thus on the other hand, let. The following standard lemma is often useful, and makes explicit a bit of intuition.
A quasimetric is a distance function which satisfies the triangle inequality but is not symmetric. Reconciling domains with metric spaces, lectures notes in. However,the converse is not true as clear from the following example. Uniform metric let be any set and let define particular cases. Hardy spaces on ahlforsregular quasi metric spaces a sharp. Our motivation comes from biological sequence comparison.
We provide examples to assure the validity of the given results. It is clear that b metric spaces, quasib metric spaces and b metriclike spaces are dqb metric spaces but converse is not true. A fixed point theorem for generalized contractions. A metric space is complete if every cauchy sequence has a limit. A sequence x, in the quasi metric space x, d is called cauchy sequence provided that for any natural number k there exist a yk e x and an nk. On hybrid contractions in the context of quasimetric spaces. Pdf the purpose of this work is to study topological properties of bdislocated quasimetric space. Traditionally this is done by generalizing the concept of cauchy sequence andor that of the convergence of a sequence. This paper contains a study of families of quasipseudometrics the concept of a quasipseudometric was introduced by wilson 22, albert 1 and kelly 9 generated by proba bilistic quasipseudometricspaces which are generalization of probabilistic metric space pmspace shortly 2, 3, 4, 6.
Completely metrizable spaces are often called topologically complete. Nearly all the concepts we discuss for metric spaces are natural generalizations of the corresponding concepts for r with this absolutevalue metric. A particular case of the previous result, the case r 0, is that in every metric space singleton sets are closed. We show that the bicompletion of a weightable quasimetric space is a weightable quasimetric space. Every extended metric can be transformed to a finite metric such that the metric spaces are equivalent as far as notions of topology such as continuity or convergence are concerned. It is evident that any metric space is a quasi metric space, but the converse is not true in general. In this manuscript, we will investigate the existence of fixed points for mappings that satisfy some hybrid type contraction conditions in the setting of quasimetric spaces. The tools established in the first part are then used to develop a sharp theory of besov and triebellizorkin spaces in ahlforsregular quasi metric spaces. Topology and its applications 30 1988 127148 127 northholland on completeness in quasimetric spaces doitchin doitchinov department of mathematics, university of sofia, 1090 sofia, bulgaria received 5 september 1986 revised 24 august 1987 a notion of cauchy sequence in quasimetric spaces is introduced and used to define a standard completion for a special class of spaces. Pdf contribution of fixed point theorem in quasi metric.
Since the conclusion of the baire category theorem is purely topological, it applies to these spaces as well. U nofthem, the cartesian product of u with itself n times. Hans triebel a new approach to function spaces on quasimetric spaces where. Quasi metric spaces the basic theory outline 1 introduction 2 the basic theory 3 transition systems 4 the theory of quasi metric spaces 5 completeness 6 formal balls 7 the quasi metric space of formal balls 8 notions of completion 9 conclusion. Lectures on quasiisometric rigidity michael kapovich june 28, 2012. Bicompleting weightable quasimetric spaces and partial. We remark that semantics for rewrite systems typically involve free universal. Dislocated quasibmetric spaces and fixed point theorems. Let be a quasi metric space and let be a sequence in and. On completeness in quasimetric spaces sciencedirect.
Families of quasi pseudometrics generated by probabilistic quasi pseudo metric spaces mariusz t. But it is not dislocated quasi b metric space nor bmetric space. Department of mathematics, university of leicester, leicester, lei 7rh. In calculus on r, a fundamental role is played by those subsets of r which are intervals. Class of fundamental groups of closed compact, without boundary surfaces. Oct 12, 2017 the next result can be stated when one considers a quasi metric space, a particular instance of partial quasi metric spaces, in the statement of theorem 6.
Let xr and suppose dx, y then x, d is a dislocated quasi bmetric space with the coefficient k2. In particular, applications to two interesting examples of partial metric spaces which appear in computer science, as the domain of. In this paper, we obtain some fixed point theorems for fuzzy mappings in a left ksequentially complete quasi pseudo metric space and in a right ksequentially complete quasi pseudo metric space, respectively. The text is divided into two main parts, with the first part providing atomic, molecular, and grand maximal function. Our analysis is based on the fact that fuzzy fixed point results can be obtained from the fixed point theorem of multivalued mappings with closed values. Quasimetric spaces and pointfree geometry article pdf available in mathematical structures in computer science 161. In this paper we shall obtain a fixed point theorem for generalized contractions with respect to wdistances on complete quasimetric spaces from which we deduce wdistance versions of boyd and wongs fixed point theorem 17 and of matkowskis fixed point theorem 18. The above two nonstandard metric spaces show that \distance in this setting does not mean the usual notion of distance, but rather the \distance as determined by the. They called a t 0quasimetric space qhyperconvex if and only if it is injective in the category of t 0quasimetric spaces and nonexpansive maps.
Metricandtopologicalspaces university of cambridge. Quasi isometry of metric spaces is an equivalence relation. The analogues of open intervals in general metric spaces are the following. Notes on metric spaces these notes introduce the concept of a metric space, which will be an essential notion throughout this course and in others that follow. In this paper we introduce the concept of generalized quasi metricgq space and establish some fixed point theorems in gq metric spaces. We establish convergence theorems of a sequence in quasimetric spaces. Fixed point results in dislocated quasi metric spaces. We show that the bicompletion of a weightable quasi metric space is a weightable quasi metric space. Some coupled xed point theorems on quasi partial b metric spaces 297 4 the main results theorem 4. This is a cauchy sequence of rational numbers, but it does not converge towards any rational limit. The result, presented here, not only generalizes a number of existing results, but also unifies several ones on the topic in the literature.
Quasimetric tree in t0quasimetric spaces sciencedirect. Quasimetric spaces, quasimetric hyperspaces and uniform. The main purpose of this paper is to extend the convergence and cauchy conditions in a quasimetric space by. Introduction some convergence theorems of certain iterations to a. Quasimetric spaces transition systems prevision transition systems add action labels 2l, to control system. Let be a quasimetric space and let be a sequence in and. Chapter 2 presents basic concepts of the theory of. Then, the metric spaces are the quasimetric spaces satisfying the symmetric property d0 d x, y d y, x.
Completely metrizable spaces can be characterized as those spaces that can be written as an intersection of countably many open subsets of some complete metric space. A new generalized quasi metric space with some fixed. However, the free choice of s 0 allows us to take s t0. Once again, to prove that this is a metric, one needs to check the three axioms. Our main theorem extends and unifies existing results in the recent literature. We extend some of the mmspace concepts to the setting of a quasi metric space with probability measure pqspace. In particular, for a subclass of kmfuzzy quasimetric spaces, yoneda t completeness. A metric induces a topology on a set, but not all topologies can be generated by a metric. Some coupled fixed point theorems on quasipartial b. A completion of a quasi metric space x, d is a complete quasi metric space x, d in which x, d can be quasi isometrically embedded as a dense subspace. Now, we recall convergence and completeness on quasimetric spaces.
Then d is a metric on r2, called the euclidean, or. A proposal to the study of contractions in quasimetric spaces. Since quasi metric spaces form a generalization of metric spaces, any sound completion theory for such spaces should strictly generalize the usual completion theory for metric spaces. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc. Now, we recall convergence and completeness on quasi metric spaces. Systematically constructing an optimal theory, this monograph develops and explores several approaches to hardy spaces in the setting of alhlforsregular quasimetric spaces. T,quasimetric space x, d has to be defined in such a manner that the following requirements are fulfilled.
A new approach to function spaces on quasimetric spaces. Generalizing a result of schwartz, any quasiisometric image of a nonrelatively hyperbolic space in a relatively hyperbolic space is contained in a bounded. This can be done using a subadditive monotonically increasing bounded function which is zero at zero, e. In this paper, we establish dislocated quasi b metric spaces and introduce the notions of geraghty type dqbcyclicbanach contraction and dqbcyclickannan mapping and derive the existence of fixed point theorems for such spaces. The result, presented here, not only generalizes a number of existing results, but also unifies several ones on the topic in the. Quasicontractions on metric spaces 325 for all x,y in x for which the righthand side of the inequality is positive. A fixed point theorem for generalized contractions involving. Ciric, a generalization of banachs contraction principle, proc amer. In this manuscript, we will investigate the existence of fixed points for mappings that satisfy some hybrid type contraction conditions in the setting of quasi metric spaces. A sequence x, in the quasimetric space x, d is called cauchy sequence provided that for any natural number k there exist a yk e x and an nk.
Metric space, contraction mapping, fixed point theorem, quasi metric space, pconvergent, porbit ally continuous. Here we describe tysons theorem on the equivalence of geometric quasiconformality, localquasisymmetry, and a variant of metric quasiconformality on domains of metric spaces and give a summary of the main points and techniques of its proof. Let f be a monotone mapping from a smyth complete quasimetric space x, q into itself such that. Generalized contractive setvalued maps on complete. Wait m 1 m 2 init insertcoin cancel insertcoin cancel pressbutton servecoffee cashin serving served l is a set ofactionsthatphas control over. Some coupled fixed point theorems in quasipartial metric. The phenomenon of concentration of measure on high dimensional structures is usually stated in terms of a metric space with a borel measure, also called an mmspace. The results of this paper generalize several known theorems in the recent literature. A topological space whose topology can be described by a metric is called metrizable an important source of metrics in differential. On completeness in quasimetric spaces introduction core.
Then the pair x,d is called dislocated quasi bmetric space or in short dqb metric space. Pdf quasi metrics have been used in several places in the. The space q of rational numbers, with the standard metric given by the absolute value of the difference, is not complete. The central result of this thesis, developed in chapter 3, is that a natural correspondence exists between similarity measures between biological nucleotide or protein sequences and quasimetrics. Quasi contractions on metric spaces 325 for all x,y in x for which the righthand side of the inequality is positive. We extend some of the mmspace concepts to the setting of a quasimetric space with probability measure pqspace. On the yoneda completion of a quasimetric space sciencedirect.
The metric is often regarded as a distance function. In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set. Fixed point theorems for fuzzy contractive mappings in quasi. Dislocated quasi bmetric space and new common fixed point. In particular, they are independent of the euclidean charts. Class of fundamental groups of closed compact, without boundary 3.
In this paper we shall obtain a fixed point theorem for generalized contractions with respect to wdistances on complete quasi metric spaces from which we deduce wdistance versions of boyd and wongs fixed point theorem 17 and of matkowskis fixed point theorem 18. Pdf on generalized quasi metric spaces researchgate. Xthe number dx,y gives us the distance between them. In problems about completeness and completions, those quasi metric spaces with a hausdorff topology are the most important.
We also show that several distinguished examples of nonmetrizable quasimetric spaces and of cones of asymmetric normed spaces admit distances of this type. Dislocated quasibmetric spaces and fixed point theorems for. New fixed point results in partial quasimetric spaces in. The relationships between kmfuzzy quasimetric spaces and the. Informally, 3 and 4 say, respectively, that cis closed under. Recall that a quasipseudometric space is an ordered pair x,p, where x is a nonempty set and the function p. Thereafter, we investigate not only the existence, but also the uniqueness of a fixed point of such mappings in the setting of rightcomplete quasi metric spaces. Abstracta quasimetric is a distance function which satisfies the triangle inequality but is not symmetric in general.953 779 501 961 909 4 239 365 1627 1298 408 355 71 1212 806 718 312 244 667 178 533 413 1124 121 640 1482 829 1245 790 778 990 1501 264 774 437 929 945 1151 220 1319 208 7 394 51 241 1228 124 1154 1403 957