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Topic: Cauchy completion

	Cauchy sequence - Wikipedia, the free encyclopedia
		Cauchy sequences require the notion of distance so they can only be defined in a metric space.
		They are of interest because in a complete space, all such sequences converge to a limit, and one can test for the Cauchy property without knowing the value of the limit (if it exists), in contrast to the definition of convergence.
		Any Cauchy sequence with a modulus of Cauchy convergence is equivalent (in the sense used to form the completion of a metric space) to a regular Cauchy sequence; this can be proved without using any form of the axiom of choice.
	en.wikipedia.org /wiki/Cauchy_sequence (1047 words)

	PlanetMath: completion
		with this metric is of course a complete metric space.
		This is version 5 of completion, born on 2002-05-21, modified 2005-08-21.
		Object id is 2923, canonical name is Completion.
	planetmath.org /encyclopedia/Completion.html (276 words)

	Complete space - Wikipedia, the free encyclopedia
		In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M.
		Note that completeness is a property of the metric and not of the topology, meaning that a complete metric space can be homeomorphic to a non-complete one.
		Completely metrizable spaces can be characterized as those spaces which can be written as an intersection of countably many open subsets of some complete metric space.
	en.wikipedia.org /wiki/Cauchy_completion (1248 words)

	PlanetMath: value group of completion
		"value group of completion" is owned by pahio.
		Cross-references: ultrametric triangle inequality, positive, non-archimedean field, Cauchy sequence, extension, completion, multiplicative group, subgroup, value group, valuation, non-archimedean, field
		This is version 5 of value group of completion, born on 2005-01-27, modified 2005-04-26.
	planetmath.org /encyclopedia/ValueGroupOfCompletion.html (133 words)

	P-adic Numbers
		If s and t are cauchy, and s-t converges to 0, then s and t represent the same point in the completion of our metric space.
		Furthermore, the original space embeds in the completion as a subring, and as a subspace with the induced topology.
		Thus a sequence of rationals that is cauchy, using the valuation metric, represents a p-adic number.
	www.mathreference.com /id-val,padic.html (1197 words)

	Cauchy sequence - Article from FactBug.org - the fast Wikipedia mirror site
		In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses.
		They are of interest because in a complete space, all such sequences converge to a limit, and one can test for "Cauchiness" without knowing the value of the limit (if it exists), in contrast to the definition of convergence.
		), then this completion is canonical in the sense that it is isomorphic to the inverse limit of (G/H), where H varies over all normal subgroups of finite index.
	www.factbug.org /cgi-bin/a.cgi?a=6085 (598 words)

	Extending a Uniform Function
		The function 1/x is continuous on (0,1), yet the cauchy sequence 1/n does not map to a cauchy sequence in the range.
		A cauchy sequence does map to a cauchy sequence, however, when its limit point p is in the domain.
		This sequence is cauchy, and its image in t is cauchy, with a limit point y.
	www.mathreference.com /top-ms,extend.html (978 words)

	Cauchy Completeness: Complete Metric Space, Banach Space, and Hilbert Space
		The answer is this: in a finite dimensional complex vector space, a Cauchy sequence always has a limit in that vector space; in other words, a finite dimensional vector space is complete.
		The Cauchy completion of the rationals are the reals.
		The Cauchy completion of an inner product space is a Hilbert space The Cauchy completion of a normed linear space is a Banach space.
	www.math.ohio-state.edu /~gerlach/math/BVtypset/node11.html (326 words)

	The Analysis of Informatic Phenomena: Research seminars
		The Scott-topology on a continuous order and the object of subobjects of an ordered sheaf on a locale then turn out to be examples of the same categorical construction: namely that of "regular presheaves" on a regular semicategory.
		But to recover a well-behaved Cauchy completion for such regular semicategories, we must impose an even stronger "total regularity" condition.
		Finally I want to show how such Q-orders can also be described as categories enriched in the split-idempotent completion of Q: this reconciles the results of Sydneysiders and Lovanists on that subject in the case where Q is a locale.
	web.comlab.ox.ac.uk /oucl/seminars-tt04/extra/stubbe.html (250 words)

	Bethlehem University - Faculty of Science - Department of Math (Site not responding. Last check: 2007-10-09)
		Completion of MATH 141, MATH 142 and MATH 241.
		Metric spaces, convergence and continuity, completeness and Cauchy’s completion theorem, general topological spaces, separation axioms, metrizability, compactness, and connectedness, compactification theorems, product spaces and Tychonof theorem, the fundamental group and an introduction to homotopy theory are included in the course.
		Study, planning, development and operation phase structures of the life cycles of a system, planned versus achievement examination and project control, design of a complete system and possibility of implementing a part of it as a project are also covered.
	www.bethlehem.edu /programs/science/math.shtml (1665 words)

	Mathematical logic in Padua - Events - Second Workshop on Formal Topology - Tutorials
		An introduction to the notion of uniformity on a frame and various concepts connected with this such as: completeness, completion, Cauchy completeness and Cauchy completion, uniformities and compactifications, special uniformities.
		Then I will recall that it is still possible to deal with subsets, and their algebra, within type theory if quantification over them is avoided or limited to the case of set-indexed families of subsets.
		Indeed the conditions that we require on the cover relation and the positivity predicate are clearly valid in any concrete topological space but nothing guarantee for their completeness.
	www.math.unipd.it /~logic/events_2wftop_tutorials.html (1052 words)

	Atlas: Cauchy complete l-ideals of a lattice ordered group by Stefan Cernak (Site not responding. Last check: 2007-10-09)
		Vulikh has defined the notion of convergence of sequences with a regulator in a vector lattice V. A "Convergence regulator" depends on a sequence.
		The Dedekind completion D(G) of G is u-Cauchy complete (C-complete) and a u-Cauchy completion C(G) of G is an l-subgroup of D(G).
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # capw-25.
	atlas-conferences.com /cgi-bin/abstract/capw-25 (209 words)

	Elena Alemany (Site not responding. Last check: 2007-10-09)
		It is well known that every metric space has an (up to isometry) unique metric completion.
		In the non-symmetric case, it is possible to find examples of Hausdorff quasi-metric spaces which do not admit a quasi-metric completion (not even for a very general notion of it, such as half-completion.
		\rangle is a Cauchy sequence (in the sense of Fletcher and Lindgren) in (X, d) and x in X is a T(d
	www.utm.edu /staff/jschomme/topology/c/a/a/h/04.htm (295 words)

	NISC South Africa
		In this article completions of special probabilistic semiuniform convergence spaces are considered.
		It turns out that every probabilitic Cauchy space under a given t-norm T (triangular norm) has a completion which, in the special case of probabilistic Cauchy spaces with reference to T = min, coincides with the Kent-Richardson completion for probabilistic Cauchy spaces.
		Moreover, a completion of probabilistic uniform limit spaces T = min is given which in case of constant probabilistic uniform limit spaces coincides with the Wyler completion.
	www.nisc.co.za /oneAbstract?absId=619 (165 words)

	complete-1.html (Site not responding. Last check: 2007-10-09)
		The first has the virtue of being modeled on the usual proof using Cauchy sequences.
		It's downside is in its complexity since the members of the Completion are equivalence classes of Cauchy sequences.
		That the embedding is dense, is the usual argument that a Cauchy sequence is the limit of its terms.
	www.umsl.edu /~siegel/SetTheoryandTopology/complete-1.html (154 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		In terms of (an ultrametric version of) the categorical Yoneda embedding we construct 1.
		As an illustration of the use of the theory in semantics, a domain for `quantitative' simulation is constructed.
		Then the objects of set^{Sig} can be thought of as formulas or Makkai sketches and the maps as clauses or sequents.
	www.math.mcgill.ca /rags/seminar/seminar.listings.96 (884 words)

	J.J.M.M. Rutten (Site not responding. Last check: 2007-10-09)
		The completion of a generalized ultrametric space is defined along the lines of Smyth' Cauchy completion of quasi metric spaces.
		Following the approach of Lawvere in which generalized ultrametric spaces are viewed as [0,1] -enriched categories, an alternative, equivalent definition of completion is then formulated in terms of a metric version of the Yoneda Lemma, leading to simple proofs of some properties.
		Next a topology for generalized ultrametric spaces is defined, which combines both the Scott topology on preorders and the ordinary \epsilon -ball topology on ultrametric spaces, in a way that preserves the nice properties of both.
	www.utm.edu /~jschomme/topology/c/a/a/f/73.htm (226 words)

	Spreads and choice in constructive mathematics
		The former have not been shown to be Cauchy complete (without using choice), which makes them a little unsatisfactory---the reason we constructed the reals in the first place was because the rationals were not complete!
		In [3] the problem posed by the fundamental theorem of algebra is solved by redefining what a solution is. Instead of trying to approximate a single root of the polynomial, we approximate the set of roots---the spectrum.
		In particular, the completion of M itself can be considered to be an M-spread, as in the traditional intuitionistic view of the continuum R as a Q-spread.
	www.math.fau.edu /Richman/Docs/spreads.htm (2237 words)

	does .99~ = 1
		For the, what, tenth time, the real numbers are defined to be (amongst other equivalent definitions) the completion of Cauchy sequences of rationals wrt the Euclidean metric.
		There is more to maths than that which can be constructed in finite time.
		Cauchy sequences of rationals wrt hte Euclidean metric are "raping" the infinite to be finite, no more no less.
	www.physicsforums.com /showthread.php?t=17803 (671 words)

	Spring 96 Abstracts
		Access to complete papers in the volumes is available on the web to those whose home institution maintains a subscription to TCS.
		Since conference proceedings of the type just described often consist largely of "extended abstracts" rather than complete journal papers, the editors expect and encourage conferences that publish their proceedings as volumes in ENTCS also to seek publication of journal versions of some of the papers in their proceedings with a major journal.
		Topology Atlas intends to be a complete historical and living portrait of the entire topological community, its endeavors (past and present), and its accomplishments --- basically a living encyclopaedia!
	math.tntech.edu /spring-top/spring96-abstracts.html (11944 words)

	Roman Fric
		Fric, R., On the completion of sequential structures.
		Fric, R., Kent, D. C., On the natural completion functor for Cauchy spaces.
		Fric, R., Kent, D. C., On functiorial completions of Cauchy spaces.
	www.saske.sk /MI/eng/fric.htm (1129 words)

	Real Numbers as Equivalence Classes of Cauchy Convergent Sequences
		Although it is tempting, and commonly done, to define real numbers as infinite sequences of digits there are insurmountable logical difficulties with that construction.
		This shows the sum, difference, product and reciprocal are well defined.
		This approach to the real numbers uses eleven axioms to define a complete ordered field, the real numbers.
	www.sjsu.edu /faculty/watkins/cauchy.htm (1120 words)

	Change of base, Cauchy completeness and reversibility
		We investigate the effect on Cauchy complete objects of the change of base 2-functor ${\cal V}-Cat \rightarrow {\cal W}-Cat$ induced by a two-sided enrichment ${\cal V} \rightarrow {\cal W}$.
		The reversibility notion introduced by Walters is extended to two-sided enrichments and Cauchy completion.
		Keywords: Enriched categories, two-sided enrichments, change of base, reversibility, Cauchy completion, sheaves.
	www.maths.tcd.ie /EMIS/journals/TAC/volumes/10/10/10-10abs.html (115 words)

	Completeness
		A metric space is if every Cauchy sequence converges, and one can show that the completion of
		is the set of equivalence classes of Cauchy sequences, and there is a natural injective map from
		are continuous, they induce well-defined field operations on equivalence classes of Cauchy sequences componentwise.
	modular.fas.harvard.edu /papers/ant/html/node63.html (204 words)

	week199
		In topology, a "spectrum" is defined to be a sequence of pointed topological spaces, each of which is homeomorphic to the space of all based loops in the next.
		If you start with a braided monoidal category, the group completion of its classifying space will be a double loop space.
		In particular, the Cauchy completion procedure is possible to formulate in purely algebraic terms.
	math.ucr.edu /home/baez/week199.html (4511 words)

	A glimpse at p-adic numbers (Site not responding. Last check: 2007-10-09)
		It reviews how to get to the reals via the rationals and generalizes to rings, and then introduces a norm on the rationals where 27 is small and 81 is smaller if p=3.
		It carries out the Cauchy completion using this norm in place of the absolute value which Cauchy used as a norm and gets, voilá, a structure isomorphic to that from the book.
		Matthew Watkins has an interesting approach to p-adic numbers along these lines.
	www.cap-lore.com /MathPhys/p-adic.html (426 words)

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