Paper 2, section i 4e metric and topological spaces. Metricandtopologicalspaces university of cambridge. Human memory as a p adic dynamic system springerlink. Metric spaces are generalizations of the real line, in which some of the theorems that hold for r. Here is a more mathematical way of saying all this. Arbitrary intersectons of open sets need not be open. Any normed vector space can be made into a metric space in a natural way.
A stream processing approach to distance measurement of integers in padic metric space. With the padic topology, z is an ultrametric space. It is well known that q equipped with the metric induced by the euclidean norm. First, however, we must develop language that we can use in constructing and describing the p adic metric. X, d is a complete metric space if any cauchy sequence in x has a limit in x.
This is what allows the development of calculus on q p, and it is the interaction of this analytic and algebraic structure that gives the p adic number systems their power and utility. Recall that a topological space is complete if every cauchy. The work of this paper will be further simplified by assuming that the process y. The p adic topology on z is the metric topology with the p adic metric d. Defn a subset c of a metric space x is called closed if its complement is open in x. Concretely, a p p adic integer x x may be written as a base p p expansion. Moreover, each o in t is called a neighborhood for each of their points. U nofthem, the cartesian product of u with itself n times. Thanks for contributing an answer to mathematics stack exchange. A set with a metric, such as d in the definition above, is called a metric space. The completion theorem 6 every metric space m, and in our context elds f, can be completed, i. Proof we prove sequential compactness since we are in metric space.
Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc. This was worked out by berthelot ber91, but is rather subtle. Maxda,b,db,c 1 a, b and c are points of this ultrametric space, instead of the usual triangular inequality, characteristic of euclidean geometry. The algorithm produces partitions of the p adic metric space having a very simple geometry. Introduction to p adic numbers an overview of ultrametric spaces and p adic numbers. The analogues of open intervals in general metric spaces are the following. On the other hand, this metric topology also is the locally convex topology given by the family of lattices. Pictures of ultrametric spaces, the padic numbers, and valued fields jan e.
We assume that two ideas are close if they have a sufficiently long initial segment in common. Pictures of ultrametric spaces, the padic numbers, and. Each of the following is an example of a closed set. Each has a unique padic expansion with no negative powers of p. A metric space is a set xtogether with a metric don it, and we will use the notation x. If v,k k is a normed vector space, then the condition du,v ku. In contrast, the p adic extension arises from the use of the counterintuitive p adic metric. An ultrametric or nonarchimedean metric on a set x is function d. This metric on mg,r is called the weilpetersson metric. Eichlinghofen, the 28th august 2015 by gilles bellot tudortmunduniversity facultyofmathematics. Y,d y is a metric space and open subsets of y are just the intersections with y of open subsets of x. Introduction to padic numbers an overview of ultrametric spaces and padic numbers. In a complete metric space, a sequence is convergent if and only if it is a cauchy.
We propose a mathematical model of the human memoryretrieval process based on dynamic systems over a metric space ofpadic numbers. Ultrametricity an ultrametric space is a space endowed with an ultrametric distance, defined as a distance satisfying the inequality da,c. Let a be a dense subset of x and let f be a uniformly continuous from a into y. Each compact metric space is complete, but the converse is false. The padic numbers are most simply a field extension of q, the rational.
A metric space x, d is a topological space, with the topology being induced from the metric d. We also assume that this dynamic system is located in the subconscious and is. The p adic integers form a subset of the set of all p adic numbers. With the p adic topology, z is an ultrametric space. Integers that are congruent modulo a high power of p have a di erence with a large padic valuation, and hence are assigned a. Whereas in the adic world, there is a formal unit disc bered over a twopoint space spaz p, and its generic ber is simply the open. This metrics allowed for an implicit use of human visual. The padic integers form a subset of the set of all padic numbers.
Pictures of ultrametric spaces, the p adic numbers, and valued fields jan e. The latter in turn constitute an extension of the eld of rational numbers, analogous to the completion of the rationals by the real numbers with respect to the standard ordinary metric. In this work we gave some definitions and properties about both metric and ultra metric norms. A subset is called net if a metric space is called totally bounded if finite net. Further, a metric space is compact if and only if each realvalued continuous function on it is bounded and attains its least and greatest values. Most crucial property of this norm is that it satisfies the ultra metric triangle inequality. A metric space consists of a set xtogether with a function d. Igusas theorem on the rationality of the zeta function 15 5. Note that because of the division problem, when p is not prime, the padic numbers are not a field, but only a ring. A rather trivial example of a metric on any set x is the discrete metric dx,y 0 if x. A metric space is a set with a distance function or metric, dp, q defined over the. In jerzy browkins 1978 and 2000 papers on padic continued fractions, several algorithms for computing continued frac. Pdf a stream processing approach to distance measurement of.
In calculus on r, a fundamental role is played by those subsets of r which are intervals. The completion theorem describes the way in which a metric space can be completed and what completion means. First, however, we must develop language that we can use in constructing and describing the padic metric. A crucial difference is that the reals form an archimedean field, while the p padic numbers form a nonarchimedean field. When the space is z, q,orr, we usually form such a picture by imagining points on the number line. Before delving into the connection between the collatz conjecture and p adic numbers, we must rigorously introduce the ring of p adic integers.
The completion of q with respect to the padic metric is denoted by qp and is called the field of. This metric space is complete in the sense that every cauchy sequence converges to a point in q p. Note that iff if then so thus on the other hand, let. One norm that we are quite familiar with is the absolute value, jj. When studying a metric space, it is valuable to have a mental picture that displays distance accurately. Integers that are congruent modulo a high power of p have a di erence with a large p adic valuation, and hence are assigned a. If p is prime then the padic numbers form a complete metric space containing the rational numbers it is a completion of the rational numbers and it is also a field. D such that, 1 m is complete with respect to the metric d. But avoid asking for help, clarification, or responding to other answers. Then we call k k a norm and say that v,k k is a normed vector space.
In contrast, the padic extension arises from the use of the counterintuitive padic metric. Pdf segmentation of images in padic and euclidean metrics. V w between banach spaces is continuous for the norm topologies if and only if it is bounded i. If v is a normed space which is complete under its norm or rather, under the induced ultrametric dv,w kv. This makes us wonder if the ring of padic numbers with their padic soft metric space is a s. The goal of these notes is to construct a complete metric space. If a subset of a metric space is not closed, this subset can not be sequentially compact. The padic completion of q and hensels lemma contents. X a, there is a sequence x n in a which converges to x. A metric space is, essentially, a set of points together with a rule for saying how far apart two such points are.
In a complete metric space, a sequence is convergent if and only if it is a cauchy sequence. Metric spaces joseph muscat2003 last revised may 2009 a revised and expanded version of these notes are now published by springer. Itisknownthat the canonical coordinates associated to the weilpetersson metric coincide with the socalled bers coordinates on m g,r the universal covering space of mg,r. This is a null sequence since for big enough n, all x n are zero. I highly recommend ne chapter ii for a detailed discussion of this topic. Often, if the metric dis clear from context, we will simply denote the metric space x.