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