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# Topic: Totally bounded

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 Encyclopedia: Totally bounded   (Site not responding. Last check: 2007-10-21) Since the set of the centres of these balls is finite, it has finite diameter, from which it follows (using the triangle inequality) that every totally bounded space is bounded. A useful characterisation of compactness for metric spaces is that a metric space is compact if and only if it is complete and totally bounded. a bounded set is referred to as "a finite interval" or "finite region". www.nationmaster.com /encyclopedia/Totally-bounded   (1684 words)

 Totally bounded space -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-10-21) One can also define totally bounded spaces directly, and then define a set to be totally bounded if and only if it is totally bounded when considered as a (A space that is contained within another space) subspace. There is a nice relationship between total boundedness and (The consistency of a compact solid) compactness: A space is compact (Click link for more info and facts about if and only if) if and only if it is both totally bounded and (Click link for more info and facts about Cauchy complete) Cauchy complete. But it is no longer true (that is, the proof requires choice) that every totally bounded space is precompact; in other words, the completion of a totally bounded space might not be compact in the absence of choice. www.absoluteastronomy.com /encyclopedia/t/to/totally_bounded_space.htm   (1019 words)

 Interaction - Smart Board A "totally bounded" set is one that can be "covered" by a finite number of open balls (think of them as "disks") of any given diameter. To prove the assertion it is sufficient to show that any infinite sequence in a totally bounded set must contain a Cauchy sequence (and thus its closure is compact). One implication of total boundedness is that we may obtain an open cover consisting of a finite number of e-balls, represented iconically as circles. www.cs.brown.edu /stc/resea/interaction/research_I12a.html   (974 words)

 Heine–Borel theorem - Wikipedia, the free encyclopedia A subset of a metric space is compact if and only if it is complete and totally bounded. A similar case holds for unbounded sets with finite covers: the elements of the finite cover could not possibly be bound in size, otherwise the union of all the elements of the finite cover would itself be bounded, and could not cover an unbounded set. It is obvious that any compact set E is totally bounded. en.wikipedia.org /wiki/Heine-Borel_theorem   (1040 words)

 [No title] Thanks for enlightening me. Keep in mind that "bounded" only makes sense in a metric space, and that "closed" implies X is a subset of something else (evey topological space is closed in itself). Here, a space is "totally bounded" if for every e>0 X can be covered with finitely many balls of radius e. Totally bounded (or "pre-compact") means that for every positive epsilon there exists a finite epsilon-dense subset (or equiv., there exists a finite cover by sets of diameter < epsilon). www.math.niu.edu /~rusin/known-math/95/compact.nss   (1036 words)

 PlanetMath: bounded is said to be bounded if there is some positive real number This is version 5 of bounded, born on 2002-01-01, modified 2003-11-12. Object id is 1167, canonical name is Bounded. planetmath.org /encyclopedia/Bounded.html   (62 words)

 Math218 Give an example of a bounded functional on L^\infinity which cannot be represented via an L^1 function (ie show that the dual of L^\infinity is not L^1). Note: the norm on NBV[a,b] is defined as the total variation of the function on [a,b] -- verify that this is indeed a norm. Characterization of bounded positive functionals on C_0(X) and their correspondence to finite Radon measures on X. Explanation (see first two paragraphs of Section 7.3). math.vanderbilt.edu /~neamtu/330a/topics.html   (1558 words)

 Metric space - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-10-21) A metric space M is called bounded if there exists some number r > 0 such that d(x,y) ≤ r for all x and y in M (not to be confused with "finite", which refers to the number of elements, not to how far the set extends; finiteness implies boundedness, but not conversely). It is not difficult to see that every totally bounded space is bounded. It can be shown that a metric space is compact if and only if it is complete and totally bounded. xahlee.org /_p/wiki/Metric_spaces.html   (1390 words)

 Metric space Article, Metricspace Information   (Site not responding. Last check: 2007-10-21) A metric space M is called bounded if there exists some number r, such thatd(x,y) ≤ r for all x and y in M. Since the set of the centres of these balls is finite, it has finite diameter, from which it follows(using the triangle inequality) that every totally bounded space is bounded. A usefulcharacterisation of compactness for metric spaces is that a metric spaceis compact if and only if it is complete and totally bounded. www.anoca.org /spaces/set/metric_space.html   (1572 words)

 CUNCTIPOTENT CUNCTATOR   (Site not responding. Last check: 2007-10-21) Another possible interpretation is that the total amount of space a cop can guard is meant to correspond to an open set, not necessarily an open ball. A set is totally bounded if for each epsilon>0 there exists a finite number of open balls of radius epsilon which cover the set. Certainly, the interesting thing about the city being totally bounded is that not only can it be guarded by some collection of nearsighted cops, but that it can be guarded by any collection of nearsighted cops whose nearsightedness is bounded below. cunctipotentcunctator.blogspot.com /2005_02_01_cunctipotentcunctator_archive.html   (2618 words)

 quadmax()   (Site not responding. Last check: 2007-10-21) This can substantially decrease the computational time when you know the problem is bounded. This is the case, for example, in a problem where the range of the x's is totally bounded by inequality constraints, or in a problem where x'Ax is known to have a unique maximum (A has all negative eigenvalues). However, ckbounds:F should not be used when you do not know there is a bounded solution. www.stat.umn.edu /macanova/html_help/node423.htm   (407 words)

 Bounded, Diameter   (Site not responding. Last check: 2007-10-21) The diameter of s is the least upper bound of x,y A set is totally bounded if it has a finite cover of If the set is covered by k balls of radius 1, find the ball farthest from the origin; the far edge of that ball is the bound for the set. www.mathreference.com /top-ms,bound.html   (273 words)

 Metric space If such a space is complete, we call it a Banach space. A topological space which can arise in this way from a metric space is called a metrizable space; see the article on metrization theorems for further details. The space M is called totally bounded if for every r > 0 there exist finitely many open balls of radius r whose union equals M. usapedia.com /m/metric-space.html   (1196 words)

 Atlas: Cardinal functions related to quasi-uniformities by Attila Losonczi   (Site not responding. Last check: 2007-10-21) The set of real numbers with the standard topology is also examined and it is proved that the cardinality of the set of compatible quasi-uniformities is equal to 2 and the cardinality of the transitive totally bounded quasi-uniformities is the same. 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 # cabc-28. atlas-conferences.com /cgi-bin/abstract/cabc-28   (287 words)

 totally - OneLook Dictionary Search Totally : Online Plain Text English Dictionary [home, info] Phrases that include totally: totally ordered set, totally disconnected, totally disconnected space, totally ordered, totally bounded, more... Words similar to totally: all, altogether, completely, entirely, whole, wholly, more... www.onelook.com /?w=totally&ls=a   (201 words)

 Romantic Island Hideaway, Romantic Beach, Romantic Beach Getaway, Romantic Beach Vacation The island is blessed with 14 white, naturally occurring sandy beaches, 8 of which are available for private picnics. Some are adjacent to the living and dining areas, and some are totally isolated, bounded by high cliffs or set within a cove. Your time on Turtle is your own, whether you want to stroll the beaches, wade in the turquoise waters, cruise from island to island on one of our sailboats or snorkel or scuba with a Fijian guide. www.turtlefiji.com /romance/beaches.php   (186 words)

 [No title]   (Site not responding. Last check: 2007-10-21) Subject: Re: Covering with totally- bounded sets Date: Sat, 2 Sep 2000 21:44:04 -0500 Newsgroups: sci.math Summary: [missing] On 2 Sep 2000, Jose wrote: > Let (X,d) be a metric space / there exists a countable subset D of X > such that its adherence is X. Question: > There exist Borel sets C_n n=1,2,... In an infinite-dimensional Hilbert space, each totally bounded set is nowhere dense. By the Baire category theorem, countably many such sets cannot cover the space. www.math.niu.edu /~rusin/known-math/00_incoming/tot_bdd   (152 words)

 Duality of totally bounded Abelian groups by S. U. Raczkowski and F. Javier Trigos-Arrieta   (Site not responding. Last check: 2007-10-21) Duality of totally bounded Abelian groups by S. Raczkowski and F. Javier Trigos-Arrieta Let G be a totally bounded Abelian (Hausdorff) group and denote by [G\tilde] the group of characters, i.e. The authors have granted their consent to include this document in Topology Atlas. at.yorku.ca /i/d/e/b/23.htm   (172 words)

 Volume 24, Number 1, 1998 Arhangel'skii (the main result in [1]): if a (countably) compact space X is cleavable over the class of sequential spaces, then X is also sequential. We present the original proof, based on the Doitchinov completion, that a totally bounded quiet quasi-uniformity is a uniformity. In particular it follows from Künzi's [8] proofs that each totally bounded locally quiet quasi-uniform space is uniform, and recently Déak [10] observed that even each totally bounded Cauchy quasi-uniformity is a uniformity. www.math.bas.bg /~serdica/n1_98.html   (1107 words)

 Papers reviewed by Helge Glöckner Comfort, W W and D Dikranjan, On the poset of totally dense subgroups of compact groups, Topology Proc. Möller, R G, FC^- elements in totally disconnected groups and automorphisms of infinite graphs, Math. Willis, G A, The number of prime factors of the scale function on a compactly generated group is finite, Bull. www.mathematik.tu-darmstadt.de /~gloeckner/mrlist.shtml   (597 words)

 Abstract   (Site not responding. Last check: 2007-10-21) Abstract: The aim of this note is to prove that two uniform frames, with countable bases and the same underlying frame, are equal whenever they have the same totally bounded coreflection. As corollaries we get the localic counterparts of two well-known theorems of Efremovic for uniform spaces. Keywords: uniform space, uniform frame, Weil entourage, Weil uniform frame, Samuel compactification, totally bounded coreflection. www.mat.uc.pt /~picado/publicat/9701.html   (71 words)

 MA522 Fall 2004 Homework   (Site not responding. Last check: 2007-10-21) A totally ordered set with neither a maximal nor a minimal element Prove that if X is an infinite set with the discrete metric, then X is bounded but not totally bounded. Let Y be a separable metric space and let Q⊂Y be a countable dense subset. people.clarkson.edu /~pschultz/fall04/ma522/homework.html   (209 words)

 Index of Mathematical Analysis I - The Trillia Group totally bounded sets in a metric space, 188 Greatest lower bound (glb) of a set in an ordered field, 38 Least upper bound (lub) of a set in an ordered field, 37 www.trillia.com /zakon-analysisI-index.html   (791 words)

 96-12: k-Weak Orders: Recognition and a Tolerance Result   (Site not responding. Last check: 2007-10-21) For each $k$, we give a polynomial-time recognition algorithm for $k$-weak orders and a partial characterization. In addition, we prove that among 1-weak orders, the classes of bounded bitolerance orders and totally bounded bitolerance orders are equal. This enables us to recognize the class of totally bounded bitolerance orders for 1-weak orders. dimacs.rutgers.edu /TechnicalReports/abstracts/1996.sav/96-12.html   (83 words)

 Salvador Hernández   (Site not responding. Last check: 2007-10-21) Let G be a MAPA group that is metrizable and satisfies Pontryagin duality, that is, coincides with its topological bidual. We prove that the Bohr topology of G respects compactness if and only if every non-totally bounded subset contains an infinite discrete subset which is C This result is used to characterize the Banach spaces which respect compactness (or have the Schur property, using a different terminology). www.utm.edu /~jschomme/topology/c/a/a/m/46.htm   (126 words)

 Table of contents for Library of Congress control number 2002035715   (Site not responding. Last check: 2007-10-21) The co-product topology on the algebraic direct sum 8. The embedding of totally bounded groups in pseudo-compact ones 10. The connected component of a topological group 12. www.loc.gov /catdir/toc/fy036/2002035715.html   (178 words)

 Contents of volume 49, No   (Site not responding. Last check: 2007-10-21) A Compact Convex Set not Convexly Totally Bounded A simple example of a convex compact set contained in a quasi-Banach space which is not convexly totally bounded is given. Key words: convex set, convexly totally bounded set, needle point, Ribe space. www.pan.pl /bulletin/MATH/Mat_3_01.htm   (792 words)

 [No title]   (Site not responding. Last check: 2007-10-21) We address this issue in the context of topological groups. Results are: (1) Every infinite Abelian group $G$ admits a family $\QTO{sA}{\sA}$ of $2^{2^{G}}$-many pairwise nonhomeomorphic totally bounded group topologies such that no nontrivial sequence in $G$ converges in any of the topologies $\QTO{sT}{\sT}\in\QTO{sA}{\sA}$. The speaker and co-authors will appreciate guidance on that point. www.wesleyan.edu /cgi-bin/cdf_manager/template_renderer.cgi?item=10970   (127 words)

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