We describe a completion of gmss by cauchy filters of formal balls. Also we prove a generalization of the banach contraction principle in complete generalized metric spaces. Metric spaces, generalized logic, and closed categories 3 formally in terms of three adjoint monoidal functors. Any class of spaces defined by a property possessed by all metric spaces could be called a class of generalized metric spaces. In mathematics, the concept of a generalised metric is a generalisation of that of a metric, in which the distance is not a real number but taken from an arbitrary ordered field. Reprinted in reprints in theory and applications of categories 1 2002, 7.
Bill lawvere, metric spaces, generalized logic and closed categories, rendiconti del seminario matematico e fisico di milano xliii 1973, 5166. Erik palmgren, continuity on the real line and in formal spaces. We next give a proof of the banach contraction principle in. In the last few years, the study of nonsymmetric topology has received a new derive as a consequence of its. Metric spaces, generalized logic, and closed categories the n. A metric space is, essentially, a set of points together with a rule for saying how far apart two such points are. Banachs contraction mapping principle is remarkable in its simplicity, yet it is perhaps the most widely applied fixed point theorem in all of analysis with special applications to the theory of differential and integral equations. Generalized metric spaces are a common generalization of preorders and ordinary metric spaces. The following properties of a metric space are equivalent. Steve vickers, localic completion of generalized metric spaces i, tac. If x is a topological space and x 2 x, show that there is a connected subspace k x of x so that if s is any other connected subspace containing x then s k x. Controlled metric type spaces and the related contraction. Generalized metric spaces do not have the compatible topology.
In the case of metric spaces, the compactness, the countable compactness and the sequential compactness are equivalent. Introduction when we consider properties of a reasonable function, probably the. P xof \open sets that is closed under nite intersections and arbitrary unions, meaning it satis es the following properties. A metric induces a topology on a set, but not all topologies can be generated by a metric. Citeseerx document details isaac councill, lee giles, pradeep teregowda. The equivalence between closed and boundedness and compactness is valid in nite dimensional euclidean.
We do not develop their theory in detail, and we leave the veri. A metric space is a set x where we have a notion of distance. A topological space is a set xalong with a \topology. This chapter will introduce the reader to the concept of metrics a class of functions which is regarded as generalization of the notion of distance and metric spaces. The notion of generalized metric was introduced by a branciari28 while deriving fixed point theorems for some metric like spaces. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc. New fixed point results in brectangular metric spaces. 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 metric. In this chapter we consider another type of metric called generalized metric, abbreviated g metric and study some of its topological and geometric properties. The b metric space 12, and its partial versions, which extends the metric space by modifying the triangle equality metric axiom by inserting a constant multiple s 1 to the righthand side, is one of the most applied generalizations for metric spaces see 1420.
Let x,d be a complete b metric space with constant s 1, such that b metric is a continuous functional. In general metric spaces, the boundedness is replaced by socalled total boundedness. For metric spaces, the left adjoint sends a lipschitz1 map p. Metric spaces, generalized logic, and closed categories. Localic completion of generalized metric spaces ii. In general, when we define metric space the distance function is taken to be a realvalued function. William lawvere emis the european mathematical information. In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set.
In particular, generalized metric spaces do not necessarily have the compatible topology. Following lawvere, a generalized metric space gms is a set x equipped with a metric map from x2 to the interval of upper reals. Combining lawvere s 1973 enrichedcategorical and smyths 1988, 1991 topological view on generalized metric spaces, it is shown how to construct 1 completion, 2 two topologies, and 3 powerdomains for generalized metric spaces. F in such spaces and we study their topological properties. Combining lawvere s 1973 enrichedcategorical and smyths 1988, 1991 topological view on generalized metric spaces, it is shown how to construct 1. Abstract following lawvere, a generalized metric space gms is a set x equipped with a metric map from x2 to the interval of upper reals approximated from above but not from below from 0 to. Following lawvere, a generalized metric space gms is a set x equipped with a metric map from x 2 to the interval of upper reals approximated from above but not from below from 0 to. When we prove theorems about these concepts, they automatically hold in all metric spaces, saving us the labor of having to prove them over and over again each time we introduce a new class of spaces. A generalized metric space and related fixed point theorems. Lawvere metric spaces and uniformly continuous functions. This new concept of generalized metric spaces recover various topological spaces including standard. Informally, 3 and 4 say, respectively, that cis closed under.
Taking lawveres quantale of extended positive real numbers as base quantale, qcategories are generalised metric spaces, and. Pdf localic completion of generalized metric spaces ii. Ais a family of sets in cindexed by some index set a,then a o c. Introducing a new concept of distance on a topological. Metric spaces a metric space is a set x that has a notion of the distance dx,y between every pair of points x,y. Because the underlined space of this theorem is a metric space, the theory that developed following its publication is known as the metric fixed point theory. Turns out, these three definitions are essentially equivalent. Some modified fixed point results in fuzzy metric spaces.
The term is meant for classes that are close to metrizable spaces in some sense. Lmet c is the category of lawvere metric spaces and continuous functions. Citeseerx localic completion of generalized metric spaces i. In mathematics, a metric space is a set together with a metric on the set. Following lawvere, a generalized metric space gms is a set x equipped with a metric map from x2 to the interval of upper reals approximated from above but not from below from 0 to. Localic completion of generalized metric spaces i school of. X be a contraction having contraction constant k 20,1 such that ks generalized metric space, which we call as an extended b.
Steve vickers, localic completion of generalized metric spaces ii. Simon henry, localic metric spaces and the localic gelfand duality, arxiv. We study generalized metric spaces, which were introduced by branciari 2000. Localic completion of generalized metric spaces i steven vickers abstract. The metric is a function that defines a concept of distance between any two members of the set, which are usually called points. As for the box metric, the taxicab metric can be generalized to rnfor any n.
In other words, metric embeddings are topological embeddings if the topology associated topology to the domain distinguishes points. The metric may be used to generate a topology on the set, the metric topology, and a topological space whose topology comes from some metric is said to be metrizable. On the basis of number of variables, there are many different generalizations, such as generalized metric space by mustafa and sims, generalized fuzzy metric spaces by sun and yang, new generalized metric space called metric space by sedghi, and metric spaces by abbas et al. Xthe number dx,y gives us the distance between them. A generalization of bmetric space and some fixed point. On a new generalization of metric spaces springerlink. Metric spaces, generalized logic and closed categories. Several authors see the references cited in 19,20 proved various common. Lawvere also intended the paper to serve as an accessible introduction to enriched category theory, so it begins fairly gently with some basic. On the separation axiom in a lawvere or generalized. Metric spaces, generalized logic, and closed categories springerlink. Generalized metric spaces are a common generalization of preorders and ordinary metric spaces lawvere 1973. William lawvere ftp directory listing mount allison university.