21. An Introduction To Fibrations, Topos Theory, The Effective Topos And Modest Sets Abstract A topos is a categorical model of constructive set theory. Chapter 3 is an exposition of some basic topos theory, and explains why a topos can http://www.lfcs.inf.ed.ac.uk/reports/92/ECS-LFCS-92-208/

An introduction to fibrations, topos theory, the effective topos and modest sets Wesley Phoa Abstract: A topos is a categorical model of constructive set theory. In particular, the effective topos is the categorical `universe' of recursive mathematics. Among its objects are the modest sets , which form a set-theoretic model for polymorphism. More precisely, there is a fibration of modest sets which satisfies suitable categorical completeness properties, that make it a model for various polymorphic type theories. These lecture notes provide a reasonably thorough introduction to this body of material, aimed at theoretical computer scientists rather than topos theorists. Chapter 2 is an outline of the theory of fibrations, and sketches how they can be used to model various typed lambda-calculi. Chapter 3 is an exposition of some basic topos theory, and explains why a topos can be regarded as a model of set theory. Chapter 4 discusses the classical PER model for polymorphism, and shows how it `lives inside' a particular topos - the effective topos - as the category of modest sets. An appendix contains a full presentation of the internal language of a topos, and a map of the effective topos. Chapters 2 and 3 provide a sampler of categorical type theory and categorical logic, and should be of more general interest than Chapter 4. They can be read more or less independently of each other; a connection is made at the end of Chapter 3.

22. Topos Theory - Definition Of Topos Theory In Encyclopedia For discussion of topoi in literary theory, see literary topos.In mathematics, a topos (plural topoi or toposes this is a contentious topic) is a type of http://encyclopedia.laborlawtalk.com/Topos_theory

Add to Favorites General Encyclopedia Legal ... Law forum Search Word: Visit our Law forums Labor Law Accidents Law Bankruptcy Law Business Law ... Tax Law For discussion of topoi in literary theory , see literary topos In mathematics , a topos (plural: topoi or toposes - this is a contentious topic) is a type of category which allows the formulation of all of mathematics inside it. Contents showTocToggle("show","hide") 1 Introduction 2 History 3 Formal definition 4 Further examples ... 5 References Introduction Traditionally, mathematics is built on set theory , and all objects studied in mathematics are ultimately sets and functions . It has been argued that category theory could provide a better foundation for mathematics. By analyzing precisely which properties of the category of sets and functions are needed to express mathematics, one arrives at the definition of topoi, and one can then formulate mathematics inside any topos. Of course, the category of sets forms a topos, but that is boring. In more interesting topoi, the axiom of choice may no longer be valid, or the

23. Categorical Logic - Definition Of Categorical Logic In Encyclopedia Categorical logic is a branch of category theory within mathematics, topos theory, as would now be understood, is the intuitionistic replacement for set http://encyclopedia.laborlawtalk.com/Categorical_logic

Add to Favorites General Encyclopedia Legal ... Law forum Search Word: Visit our Law forums Labor Law Accidents Law Bankruptcy Law Business Law ... Tax Law Categorical logic is a branch of category theory within mathematics , adjacent to mathematical logic but in fact more notable for its connections to theoretical computer science . In broad terms, it is a theory about the transition from a type theory , understood to be within an intuitionistic logic or constructive mathematics setting, to a category , by means of a translation that respects both the syntax and the intended computational meaning of type-theoretic constructions. The subject has been recognisable in these terms since about 1970, when the needs of domain theory started to call on category theory. The earlier history is relatively complex, and contains some ironies. Categorical logic originated within sheaf theory , as a suitable version of Kripke semantics one can say with hindsight, and emerged as a theory with a character of its own only in shedding the necessary connection with sheaves. This can be traced in a number of stages, from 1960 onwards: the formulation of the Grothendieck topos , and then of the elementary topos , giving rise first to topos theory . Topos theory, as would now be understood, is the intuitionistic replacement for

24. Re: Topos Theory For Physicists Unfortunately, quantum field theorists are likely to find topos theory a little *less* relevant to their work than other aspects of category theory, http://www.lns.cornell.edu/spr/2001-01/msg0030351.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index Re: Topos theory for physicists Subject : Re: Topos theory for physicists From : baez@galaxy.ucr.edu (John Baez) Date : Mon, 1 Jan 2001 05:42:18 GMT Approved : mmcirvin@world.std.com (sci.physics.research) Newsgroups : sci.physics.research Organization : UCR References Sender : mmcirvin@world.std.com (Matthew J McIrvin) Prev by Date: Re: A Philosophical Query and Comment about Quantizing Gravity Next by Date: Re: Hopf algebras, quantum groups Prev by thread: Re: A Philosophical Query and Comment about Quantizing Gravity Next by thread: Re: Topos theory for physicists Index(es): Date Thread

25. Re: Category Theory <-> Lambda Calculus? You d probably be interested in reading about topos theory and its relation to Anyway, anyone interested in learning a bit about topos theory, logic, http://www.lns.cornell.edu/spr/2001-03/msg0031574.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index Subject From Date : Mon, 5 Mar 2001 03:49:20 GMT Approved : mmcirvin@world.std.com (sci.physics.research) In-Reply-To 97o9ro$b5016@rs04.hrz.uni-essen.de Newsgroups : sci.physics.research Organization : University of Washington References 97o9ro$b5016@rs04.hrz.uni-essen.de Sender : mmcirvin@world.std.com (Matthew J McIrvin) http://xxx.lanl.gov/abs/gr-qc/0102108 http://xxx.lanl.gov/abs/gr-qc/0102097 http://xxx.lanl.gov/abs/gr-qc/9909056 (I can't remember which of the above listed books explains the topos theory take on lambda calculus, but IIRC at least one of them does.) Chris Hillman Home Page: http://www.math.washington.edu/~hillman/ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ NOTE TO WOULD-BE CORRESPONDENTS: I have installed a mail filter which deletes incoming messages not from the "*.edu" or "*.gov" domains, but also deletes messages from some bad actors whose emails happen to be in the "*.edu" domain and "passes" messages from a few friends with email addresses in other domains. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Follow-Ups From: baez@galaxy.ucr.edu (John Baez)

26. Categories: Topos Theory And Large Cardinals To categories@mta.ca; Subject categories topos theory and large cardinals; From FW Lawvere wlawvere@acsu.buffalo.edu ; Date Tue, 21 Mar 2000 163209 http://north.ecc.edu/alsani/ct99-00(8-12)/msg00128.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index categories: Topos theory and large cardinals To categories@mta.ca Subject : categories: Topos theory and large cardinals From wlawvere@acsu.buffalo.edu Date : Tue, 21 Mar 2000 16:32:09 -0500 (EST) Reply-To wlawvere@acsu.buffalo.edu Sender cat-dist@mta.ca http://www.acsu.buffalo.edu/~wlawvere Prev by Date: categories: Preprint: a paper on injective hulls Next by Date: categories: Functorial injective hulls Prev by thread: categories: Topos theory and large cardinals Next by thread: categories: dom fibration Index(es): Date Thread

27. Categories: Topos Theory And Large Cardinals To categories@mta.ca; Subject categories topos theory and large cardinals; From Andrej. ``Large cardinals are to ZFC, as are to topos theory. http://north.ecc.edu/alsani/ct99-00(8-12)/msg00117.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index categories: Topos theory and large cardinals To categories@mta.ca Subject : categories: Topos theory and large cardinals From Andrej.Bauer@cs.cmu.edu Date : 01 Mar 2000 22:29:58 -0500 Sender cat-dist@mta.ca Source-Info : Sender is really andrej+@gs2.sp.cs.cmu.edu User-Agent : Gnus/5.0803 (Gnus v5.8.3) XEmacs/20.4 (Emerald) Can you complete this analogy? ``Large cardinals are to ZFC, as are to topos theory.'' One answer is "Grothendieck universes", but they correspond to rather small large cardinals. Can we go further than that? Andrej Bauer School of Computer Science Carnegie Mellon University http://andrej.com Prev by Date: categories: Johnstone review of Clark/Davey Next by Date: categories: dom fibration Prev by thread: categories: Johnstone review of Clark/Davey Next by thread: categories: Topos theory and large cardinals Index(es): Date Thread

28. Topos Theory And Constructive Logic Papers Of Andreas R. Blass topos theory and Constructive Logic Papers. Andreas Blass. Papers on linear logic are on a separate page. An induction principle and pigeonhole principle http://www.math.lsa.umich.edu/~ablass/cat.html

Topos Theory and Constructive Logic Papers Andreas Blass Papers on linear logic are on a separate page An induction principle and pigeonhole principle for K-finite sets (J. Symbolic Logic 59 (1995) 11861193) PostScript or PDF We establish a course-of-values induction principle for K-finite sets in intuitionistic type theory. Using this principle, we prove a pigeonhole principle conjectured by Benabou and Loiseau. We also comment on some variants of this pigeonhole principle. Seven trees in one (J. Pure Appl. Alg. 103 (1995) 1-21) PostScript or PDF Following a remark of Lawvere, we explicitly exhibit a particularly elementary bijection between the set T of finite binary trees and the set T^7 of seven-tuples of such trees. "Particularly elementary" means that the application of the bijection to a seven-tuple of trees involves case distinctions only down to a fixed depth (namely four) in the given seven-tuple. We clarify how this and similar bijections are related to the free commutative semiring on one generator X subject to X=1+X^2. Finally, our main theorem is that the existence of particularly elementary bijections can be deduced from the provable existence, in intuitionistic type theory, of any bijections at all. Topoi and Computation (Bull. European Assoc. Theoret. Comp. Sci. 36 (1988) 57-65)

29. PhilSci Archive - Topos Theory As A Framework For Partial Truth Butterfield, Jeremy (2000) topos theory as a Framework for Partial Truth. Keywords, topos theory, category theory, partial truth, KochenSpecker http://philsci-archive.pitt.edu/archive/00000192/

About Browse Search Register ... Help Topos Theory as a Framework for Partial Truth Butterfield, Jeremy (2000) Topos Theory as a Framework for Partial Truth. Full text available as: PDF - Requires a viewer, such as Adobe Acrobat Reader or other PDF viewer. Postscript - Requires a viewer, such as GhostView - GSView Abstract This paper develops some ideas from previous work (coauthored, mostly with C.J.Isham). In that work, the main proposal is to assign as the value of a physical quantity in quantum theory (or classical physics), not a real number, but a certain kind of set (a sieve) of quantities that are functions of the given quantity. The motivation was in part physical-such a valuation illuminates the Kochen-Specker theorem; in part mathematical-the valuations arise naturally in the theory of presheaves; and in part conceptual-the valuations arise from applying to propositions about the values of physical quantities some general axioms governing partial truth for any kind of proposition. In this paper, I give another conceptual motivation for the proposal. I develop (in Sections 2

30. PhilSci Archive - A Topos Perspective On The Kochen-Specker Theorem: I. Quantum Keywords, KochenSpecker theorem; category theory; topos theory; generalised truth-values; presheaves. Subjects, Specific Sciences Physics Quantum http://philsci-archive.pitt.edu/archive/00001916/

About Browse Search Register ... Help A Topos Perspective on the Kochen-Specker Theorem: I. Quantum States as Generalised Valuations Isham, Chris and Butterfield, Jeremy (1998) A Topos Perspective on the Kochen-Specker Theorem: I. Quantum States as Generalised Valuations. Full text available as: PDF - Requires a viewer, such as Adobe Acrobat Reader or other PDF viewer. Abstract Any attempt to construct a realist interpretation of quantum theory founders on the Kochen-Specker theorem, which asserts the impossibility of assigning values to quantum quantities in a way that preserves functional relations between them. We construct a new type of valuation which is defined on all operators, and which respects an appropriate version of the functional composition principle. The truth-values assigned to propositions are (i) contextual; and (ii) multi-valued, where the space of contexts and the multi-valued logic for each context come naturally from the topos theory of presheaves. The first step in our theory is to demonstrate that the Kochen-Specker theorem is equivalent to the statement that a certain presheaf defined on the category of self-adjoint operators has no global elements. We then show how the use of ideas drawn from the theory of presheaves leads to the definition of a generalised valuation in quantum theory whose values are sieves of operators. In particular, we show how each quantum state leads to such a generalised valuation. A key ingredient throughout is the idea that, in a situation where no normal truth-value can be given to a proposition asserting that the value of a physical quantity A lies in a set D of real numbers , it is nevertheless possible to ascribe a partial truth-value which is determined by the set of all coarse-grained propositions that assert that some function f(A) lies in f(D), and that are true in a normal sense. The set of all such coarse-grainings forms a sieve on the category of self-adjoint operators, and is hence fundamentally related to the theory of presheave

31. Homotopical Algebraic Geometry I Topos Theory, By Bertrand Toen And Gabriele Vez Homotopical Algebraic Geometry I topos theory, by Bertrand Toen and Gabriele Vezzosi. This is the first of a series of papers devoted to the foundations of http://www.math.uiuc.edu/K-theory/0579/

Homotopical Algebraic Geometry I Topos theory, by Bertrand Toen and Gabriele Vezzosi 0579.bib (254 bytes) agmod-I-fin-web.dvi (516636 bytes) [July 2, 2002] agmod-I-fin-web.dvi.gz (185402 bytes) agmod-I-fin-web.ps.gz (345410 bytes) Bertrand Toen Gabriele Vezzosi

32. Category Theory (For more on the history of topos theory, see McLarty 1992. Johnstone, PT, 2002, Sketches of an Elephant a topos theory Compendium. Vol. http://plato.stanford.edu/entries/category-theory/

version history HOW TO CITE THIS ENTRY Stanford Encyclopedia of Philosophy A ... Z This document uses XHTML/Unicode to format the display. If you think special symbols are not displaying correctly, see our guide Displaying Special Characters last substantive content change FEB The Encyclopedia Now Needs Your Support Please Read How You Can Help Keep the Encyclopedia Free Category Theory 1. General Definitions, Examples and Applications 2. Brief Historical Sketch 3. Philosophical Significance Bibliography ... Related Entries 1. General Definitions, Examples and Applications defined as sets, category theory cannot provide a philosophically enlightening foundation for mathematics. In terms of collections, a category C can be described as a collection Ob , the objects of C , which satisfy the following conditions: For every pair a b of objects, there is a collection Mor a b ), namely, the morphisms from a to b in C (when f is a morphism from a to b , we write f a b For every triple a b and c of objects, there is a partial operation from pairs of morphisms in Mor a b ) X Mor b c ) to morphisms in Mor a c ), called the composition of morphisms in

33. Course In Topos Theory topos theory, spring term 1999. A graduate course (6 course points) in topos theory grew out of the observation that the category of sheaves over a http://www.math.uu.se/~palmgren/topos-eng.html

Topos Theory, spring term 1999 A graduate course (6 course points) in mathematical logic. Topos theory grew out of the observation that the category of sheaves over a fixed topological space forms a universe of "continuously variable sets" which obeys the laws of intuitionistic logic. These sheaf models, or Grothendieck toposes, turn out to be generalisations of Kripke and Beth models (which are fundamental for various non-classical logics) as well as Cohen's forcing models for set theory. The notion of topos was subsequently extended and given an elementary axiomatisation by Lawvere and Tierney, and shown to correspond to a certain higher order intuitionistic logic. Various logics and type theories have been given categorical characterisations, which are of importance for the mathematical foundations for programming languages. One of the most interesting aspects of toposes is that they can provide natural models of certain theories that lack classical models, viz. synthetic differential geometry. This graduate course offers an introduction to topos theory and categorical logic. In particular the following topics will be covered: Categorical logic: relation between logics, type theories and categories. Generalised topologies, including formal topologies. Sheaves. Pretoposes and toposes. Beth-Kripke-Joyal semantics. Boolean toposes and Cohen forcing. Barr's theorem and Diaconescu covers. Geometric morphisms. Classifying toposes. Sheaf models of infinitesimal analysis.

34. Topos -- Facts, Info, And Encyclopedia Article Background and genesis of topos theory) Background and genesis of topos theory The historical origin of topos theory is (Click link for more info and http://www.absoluteastronomy.com/encyclopedia/t/to/topos.htm

Topos [Categories: Sheaf theory, Categorical logic] For discussion of topoi in (Click link for more info and facts about literary theory) literary theory , see (Click link for more info and facts about literary topos) literary topos For discussion of topoi in (Click link for more info and facts about rhetorical invention) rhetorical invention , see (Click link for more info and facts about Inventio) Inventio In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics , a topos (plural: topoi or toposes - this is a contentious topic) is a type of (A general concept that marks divisions or coordinations in a conceptual scheme) category which allows the formulation of all of mathematics inside it. Introduction A traditional axiomatic foundation of mathematics is (The branch of pure mathematics that deals with the nature and relations of sets) set theory , in which all mathematical objects are ultimately represented by sets (even (A mathematical relation such that each element of one set is associated with at least one element of another set) functions which map between sets.) Recent work in

35. Background And Genesis Of Topos Theory -- Facts, Info, And Encyclopedia Article Lawvere s definition picks out the central role in topos theory of the (Click link for Perhaps this is why topos theory has been treated as an oddity; http://www.absoluteastronomy.com/encyclopedia/b/ba/background_and_genesis_of_top

Background and genesis of topos theory [Categories: History of mathematics, Sheaf theory, Category theory] This page gives some very general background to the mathematical idea of (A traditional theme or motif or literary convention) topos . This is an aspect of (Click link for more info and facts about category theory) category theory , and has a reputation for being abstruse. The level of abstraction involved cannot be reduced beyond a certain point; but on the other hand context can be given. This is partly in terms of historical development, but also to some extent an explanation of differing attitudes to category theory. In the school of Grothendieck During the latter part of the (The decade from 1950 to 1959) , the foundations of (Click link for more info and facts about algebraic geometry) algebraic geometry were being rewritten; and it is here that the origins of the topos concept are to be found. At that time the (Click link for more info and facts about Weil conjectures) Weil conjectures were an outstanding motivation to research. As we now know, the route towards their proof, and other advances, lay in the construction of (Click link for more info and facts about étale cohomology) étale cohomology With the benefit of hindsight, it can be said that algebraic geometry had been wrestling with two problems, for a long time. The first was to do with its

36. Algebraic Set Theory Preliminary version Relating topos theory and set theory via categories of classes Notes for lectures given at the Summer School on topos theory, http://www.phil.cmu.edu/projects/ast/

Algebraic Set Theory Algebraic set theory uses the methods of category theory to study elementary set theory. The purpose of this website is to link together current research in algebraic set theory and make it easily available. It is hoped that this will encourage and facilitate further development of the subject. Why do you call it "algebraic set theory"? Researchers in Algebraic Set Theory Steve Awodey (Carnegie Mellon University, Pittsburgh) Benno van den Berg (Utrecht Univerity) Carsten Butz (IT University, Copenhagen) Henrik Forssell (Carnegie Mellon University, Pittsburgh) Nicola Gambino (University of Cambridge) Claire Kouwenhoven-Gentil (Utrecht University) F. William Lawvere (University at Buffalo, SUNY) Ieke Moerdijk (Utrecht University) Erik Palmgren (Uppsala University) Ivar Rummelhoff (University of Oslo) Dana Scott (Carnegie Mellon University, Pittsburgh) Alex Simpson (University of Edinburgh) Thomas Streicher (Darmstadt University of Technology) Michael A. Warren

37. J. Funk's Homepage ``Rough sets and topos theory, FMCS , Ottawa, May 30 June 1, 2003. McGill Category Theory Seminar ``Purely Skeletal Geometric Morphisms II, Nov. http://www.math.uregina.ca/~funk/main.html

Jonathon Funk Department of Mathematics and Statistics University of Regina Regina, Saskatchewan email: funk@math.uregina.ca I am also Lecturer in: The Department of Computer Science, Mathematics and Physics The University of the West Indies, Cave Hill Campus Barbados email: jfunk@uwichill.edu.bb J. Koslowski's photos from Vancouver , July, 2004. That's me top right. Meetings and Conferences PAST ``The locally connected coreflection of the Jonson-Tarski topos,'' Second Saskatchewan Algebra and Number Theory Mini-Meeting , Feb. 1, 2003. ``Rough sets and topos theory,'' FMCS , Ottawa, May 30 - June 1, 2003. ``Elementary completeness,'' joint with M. Bunge, M. Jibladze, T. Streicher. 11th Union College Mathematics Meeting , Union College, Schenectady, New York. Nov. 8 and 9, 2003. ``Definable completeness,'' joint with M. Bunge, M. Jibladze, T. Streicher. 80th Peripatetic meeting on Sheaves and Logic , University of Cambridge, England, April 4, 2004. ``Rough sets,'' FMCS , June 4-6, 2004, Kananaskis, Alberta.

38. Research Areas Category Theory, Topos Theory, Topology Current It has natural connections with topos theory. Branched covers A theory of branched covers in topos theory, which is based on the ideas of Ralph Fox, http://www.math.uregina.ca/~funk/research.html

Research Areas Category theory, topos theory, topology Current Research (last updated Sept 10, 2004) Toposes and rough set theory: The idea of a rough set comes from computer science. It has natural connections with topos theory. See the Seminar of the Rough Set Technology Lab Inverse semigroups and etendue: Joint with David Cowan. Braid group orderings: Patrick Dehornoy has discovered that an Artin braid group carries a left-invariant linear ordering. In my article ``The Hurwitz action and braid group orderings,'' Theory and Applications of Categories , Vol. 9 (2001), No. 7, pp 121-150, a ramified covering space is used to find a linear ordering of a countably generated free group, which is not invariant under the product in the free group, and an action of the countably generated Artin braid group in the free group that preserves the ordering. Branched covers: A theory of branched covers in topos theory, which is based on the ideas of Ralph Fox, is developed in ``On branched covers in topos theory,'' Theory and Applications of Categories , Vol. 7 (2000), pp 1-22. This approach uses complete spread geometric morphisms.

39. Atlas: Covering Morphisms In Topos Theory By Marta Bunge In topos theory, the class of all covering projections (local homeomorphisms determined by a locally constant object of the topos) that appears explicitly http://atlas-conferences.com/c/a/j/f/21.htm

Atlas home Conferences Abstracts about Atlas Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian Categories September 23-28, 2002 Fields Institute Toronto, ON, Canada Organizers George Janelidze, Georgian Academy of Sciences, Bodo Pareigis, University of Munich, Walter Tholen, York University View Abstracts Conference Homepage Covering morphisms in topos theory by Marta Bunge Department of Mathematics and Statistics, McGill University, Montreal, Canada In the work of Janelidze in 1990, a formal notion of covering morphism arises from an abstract categorical framework given by a pair of adjoint functors. Associated with any such class of covering morphisms, there is a corresponding (pure) Galois theory, of which there are examples in different areas of mathematics. In topos theory, the class of all covering projections (local homeomorphisms determined by a locally constant object of the topos) that appears explicitly in the work of Barr and Diaconescu in 1981, has been shown already by Janelidze to be an instance of the above notion of covering morphism, but only under the conditions that either the base topos be Set (that is, for Grothendieck toposes), or else where the splitting cover in the topos is assumed connected. U (E) is defined by means of the pushout, in the 2-category Top

40. Atlas: Spectral Decomposition Of Ultrametric Spaces And Topos Theory By Alex J. Spectral Decomposition of Ultrametric Spaces and topos theory by Alex J. Lemin Moscow State University of Civil Engineering. We consider categories METR and http://atlas-conferences.com/c/a/c/l/81.htm

Atlas home Conferences Abstracts about Atlas 1999 Summer Conference on Topology and its Applications August 4-7, 1999 C.W. Post Campus of Long Island University Brookville, NY, USA Organizers Sheldon Rothman, Ralph Kopperman View Abstracts Conference Homepage Spectral Decomposition of Ultrametric Spaces and Topos Theory by Alex J. Lemin Moscow State University of Civil Engineering We consider categories METR and METR c ULTRAMETR and ULTRAMETR c of ultrametric spaces and the same maps. Given a family of ultrametric spaces, we prove that sums and products, equalizer and co-equalizer, pull-back and push-out, limits of direct and inverse spectra, if exist, are ultrametric. A product and a limit of inverse spectrum of complete metric spaces are complete. A space (X, d) is uniformly discrete for all x, y in X. This is necessarily complete. Theorem Every complete ultrametric space is isometric to a limit of a countable inverse spectrum of uniformly discrete ultrametric spaces (and vise versa) (see [1]). Corollary 1 Every compact ultrametric space is isometric to a limit of inverse sequence of skeletons of finite dimensional isosceles simplexes lying in Euclidean spaces (see [2]).

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