As a sort of converse to the above statements, the pre-image of a compact space under a proper map is compact. closely packed together. p Likewise, spheres are compact, but a sphere missing a point is not since a sequence of points can still tend to the missing point, thereby not getting arbitrarily close to any point within the space. For instance, the odd-numbered terms of the sequence 1, 1/2, 1/3, 3/4, 1/5, 5/6, 1/7, 7/8, ... get arbitrarily close to 0, while the even-numbered ones get arbitrarily close to 1. Based on the Random House Unabridged Dictionary, © Random House, Inc. 2021, Collins English Dictionary - Complete & Unabridged 2012 Digital Edition to join or pack closely together; consolidate; condense. Originally developed in 2000, by … Learn more. Definition. a formal agreement between two or more parties, states, etc. Conversely, density is the degree of compactness. ⊂ In mathematics, more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (i.e., containing all its limit points) and bounded (i.e., having all its points lie within some fixed distance of each other). A horizontal filing cabinet on rails used in offices for space efficiency expressed concisely; pithy; terse; not diffuse: (of a set) having the property that in any collection of open sets whose union contains the given set there exists a finite number of open sets whose union contains the given set; having the property that every open cover has a finite subcover. That this form of compactness holds for closed and bounded subsets of Euclidean space is known as the Heine–Borel theorem. Bolzano's proof relied on the method of bisection: the sequence was placed into an interval that was then divided into two equal parts, and a part containing infinitely many terms of the sequence was selected. a thick, bare trunk crowned by a compact mass of dark-green leaves. 13 (Metallurgy) a mass of metal prepared for sintering by cold-pressing a metal powder (C16: from Latin compactus, from compingere to put together, from com- together + pangere to fasten) Closely and firmly united or packed together; dense: compact clusters of flowers. How to use compaction in a sentence. By the same construction, every locally compact Hausdorff space X is an open dense subspace of a compact Hausdorff space having at most one point more than X. Another word for compacted. Generalisation d'un theorem de Weierstrass. For a certain class of Green's functions coming from solutions of integral equations, Schmidt had shown that a property analogous to the Arzelà–Ascoli theorem held in the sense of mean convergence—or convergence in what would later be dubbed a Hilbert space. Compactness is a "topological" property. An example of compact is making garbage or trash smaller by compressing it into a smaller mass. It was the first framework of government written and enacted in the territory that is now the United States of America, and it remained in force until 1691. Or do you just have an interest in foreign languages? We would also like a characterization of compact sets based entirely on open sets. An overview of massing in architecture. Find more ways to say compacted, along with related words, antonyms and example phrases at Thesaurus.com, the world's most trusted free thesaurus. The full significance of Bolzano's theorem, and its method of proof, would not emerge until almost 50 years later when it was rediscovered by Karl Weierstrass.[5]. The term mass is used to mean the amount of matter contained in an object. The given example sequence shows the importance of including the boundary points of the interval, since the limit points must be in the space itself — an open (or half-open) interval of the real numbers is not compact. Explore 'compact' in the dictionary. Synonym Discussion of mass. The idea of regarding functions as themselves points of a generalized space dates back to the investigations of Giulio Ascoli and Cesare Arzelà. Compact definition, joined or packed together; closely and firmly united; dense; solid: compact soil. ev This sentiment was expressed by Lebesgue (1904), who also exploited it in the development of the integral now bearing his name. Marshall Major IV wireless headphones offer great sound, plus 80+ hours of battery life and wireless charging, Jewelry organizers that will completely transform your vanity, Narrow desks that can turn any corner into a comfortable workspace. For example, an open real interval X = (0, 1) is not compact because its hyperreal extension *(0,1) contains infinitesimals, which are infinitely close to 0, which is not a point of X. That is, K is compact if for every arbitrary collection C of open subsets of X such that. 3 small, but solid and strong a short compact-looking man —compactly adverb —compactness noun [ uncountable] Examples from the Corpus compact • The apartment was ideal for the two of us - small but compact. [6] Y 1. a small cosmetics case with a mirror; to be carried in a woman's purse 2. a signed written agreement between two or more parties (nations) to perform some action 3. a small and economical car Familiarity information: COMPACT used as a noun is uncommon. In the course of the proof, he made use of a lemma that from any countable cover of the interval by smaller open intervals, it was possible to select a finite number of these that also covered it. English Collins Dictionary - English synonyms & Thesaurus. However, an open disk is not compact, because a sequence of points can tend to the boundary—without getting arbitrarily close to any point in the interior. This article incorporates material from Examples of compact spaces on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. Every topological space X is an open dense subspace of a compact space having at most one point more than X, by the Alexandroff one-point compactification. Various definitions of compactness may apply, depending on the level of generality. • COMPACT (noun) The noun COMPACT has 3 senses:. The above definition of compact sets using sequence can not be used in more abstract situations. The framework of non-standard analysis allows for the following alternative characterization of compactness:[14] a topological space X is compact if and only if every point x of the natural extension *X is infinitely close to a point x0 of X (more precisely, x is contained in the monad of x0). In 1870, Eduard Heine showed that a continuous function defined on a closed and bounded interval was in fact uniformly continuous. [13] There are pseudocompact spaces that are not compact, though. packed or put together firmly and closely The bushes grew in a compact mass. In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness. (in powder metallurgy) an object to be sintered formed of metallic or of metallic and nonmetallic powders compressed in a die. The kernel of evp is a maximal ideal, since the residue field C(X)/ker evp is the field of real numbers, by the first isomorphism theorem. Massing is the three dimensional form of a building. More example sentences. 1 (adjective) in the sense of closely packed. ; contract: the proposed economic compact between Germany and France. R It was of about 180 tons burden, and in company with the "Speedwell" sailed from Southampton on the 5th of … 1 A compact mass of a substance, especially one without a definite or regular shape. The intersection of any collection of compact subsets of a Hausdorff space is compact (and closed); A finite set endowed with any topology is compact. The uniform limit of this sequence then played precisely the same role as Bolzano's "limit point". Following the initial introduction of the concept, various equivalent notions of compactness, including sequential compactness and limit point compactness, were developed in general metric spaces. ‘After everyone had eaten, she handed them each a lump of the sticky substance.’. Either way, this quiz on Spanish words for animals is for you. Towards the beginning of the twentieth century, results similar to that of Arzelà and Ascoli began to accumulate in the area of integral equations, as investigated by David Hilbert and Erhard Schmidt. [1][2] The same set of points would not accumulate to any point of the open unit interval (0, 1); so the open unit interval is not compact. • COMPACT (adjective) denoting a tabloid-sized version of a newspaper that has traditionally been published in broadsheet form, (of a relation) having the property that for any pair of elements such that, to pack or join closely together; compress; condense, sediment compacted of three types of clay, to compress (a metal powder) to form a stable product suitable for sintering, a small flat case containing a mirror, face powder, etc, designed to be carried in a woman's handbag, a mass of metal prepared for sintering by cold-pressing a metal powder, a tabloid-sized version of a newspaper that has traditionally been publis hed in broadsheet form, Colorado joins 15 states in favor of popular vote in presidential elections. designed to be small in size and economical in operation. So Compact heat exchange is characterized by high heat transfer surface-area to volume ratios and high heat transfer coefficients compared to other exchanger types. Clump can also mean lump, like when you find a clump of grass stuck to your shoe. {\displaystyle \operatorname {ev} _{p}\colon C(X)\to \mathbf {R} } Survey. The concept of a compact space was formally introduced by Maurice Fréchet in 1906 to generalize the Bolzano–Weierstrass theorem to spaces of functions, rather than geometrical points. ) Definition 5.2.4: Open Cover : Let S be a set of real numbers. ‘Below this mass, these dense, compact objects are supported against further gravitational collapse by fermion-degeneracy pressure.’ ‘This theme is carried through to the interior with a lower seating position, aluminium trim elements, a higher centre console and a compact instrument cluster.’ all subsets have suprema and infima).[18]. What Is The Difference Between “It’s” And “Its”? Fruit should be firm and excellent in condition. Compactness, in mathematics, property of some topological spaces (a generalization of Euclidean space) that has its main use in the study of functions defined on such spaces. A non-trivial example of a compact space is the (closed) unit interval [0,1] of real numbers. Since a continuous image of a compact space is compact, the extreme value theorem: a continuous real-valued function on a nonempty compact space is bounded above and attains its supremum. We need some definitions first. For instance, some of the numbers in the sequence 1/2, 4/5, 1/3, 5/6, 1/4, 6/7, … accumulate to 0 (while others accumulate to 1). In general, for non-pseudocompact spaces there are always maximal ideals m in C(X) such that the residue field C(X)/m is a (non-Archimedean) hyperreal field. This notion is defined for more general topological spaces than Euclidean space in various ways. Freeman stands at 6 feet, 5 inches, but he’s always had a compact, whip-like swing. The Heine–Borel theorem, as the result is now known, is another special property possessed by closed and bounded sets of real numbers. 2 circumlocutory, garrulous, lengthy, long-winded, prolix, rambling, verbose, wordy. noun. : Euclidean space itself is not compact since it is not bounded. The most useful notion, which is the standard definition of the unqualified term compactness, is phrased in terms of the existence of finite families of open sets that "cover" the space in the sense that each point of the space lies in some set contained in the family. ⊂ K In the 1880s, it became clear that results similar to the Bolzano–Weierstrass theorem could be formulated for spaces of functions rather than just numbers or geometrical points. If X is a topological space then the following are equivalent: For any subset A of Euclidean space ℝn, A is compact if and only if it is closed and bounded; this is the Heine–Borel theorem. The significance of this lemma was recognized by Émile Borel (1895), and it was generalized to arbitrary collections of intervals by Pierre Cousin (1895) and Henri Lebesgue (1904). The Nursing Licensure Compact (NLC) is an agreement between states that allows nurses to have one license but the ability to practice in other states that are part of the agreement. Definition. 1, 1/2, 1/3, 3/4, 1/5, 5/6, 1/7, 7/8, ... Frechet, M. 1904. Alexandrov & Urysohn (1929) showed that the earlier version of compactness due to Fréchet, now called (relative) sequential compactness, under appropriate conditions followed from the version of compactness that was formulated in terms of the existence of finite subcovers. For any metric space (X, d), the following are equivalent (assuming countable choice): A compact metric space (X, d) also satisfies the following properties: Let X be a topological space and C(X) the ring of real continuous functions on X. Synonyms. Tell us more about your experience. 1. At the end of some of the branches come the cones, with compactly arranged and simple sporophylls all of one kind. Nursing Compact States & Nurse Licensure. [8] That is, X is compact if for every collection C of open subsets of X such that, there is a finite subset F of C such that. It is also crucial that the interval be bounded, since in the interval [0,∞), one could choose the sequence of points 0, 1, 2, 3, ..., of which no sub-sequence ultimately gets arbitrarily close to any given real number. A continuous bijection from a compact space into a Hausdorff space is a, On the other hand, the closed unit ball of the dual of a normed space is compact for the weak-* topology. The structure was so stoutly and compactly built, that four strong Indians could scarcely move it by their mightiest efforts. “Affect” vs. “Effect”: Use The Correct Word Every Time. (, This page was last edited on 30 December 2020, at 12:55. On the one hand, Bernard Bolzano (1817) had been aware that any bounded sequence of points (in the line or plane, for instance) has a subsequence that must eventually get arbitrarily close to some other point, called a limit point. Mayflower Compact, document signed on the English ship Mayflower in November 1620 prior to its landing at Plymouth, Massachusetts. Density alludes to the closeness of the atoms, in substance, i.e. [17] adj. Any finite space is trivially compact. What are Nursing Compact States? to form or make by close union or conjunction; make up or compose. Of all of the equivalent conditions, it is in practice easiest to verify that a subset is closed and bounded, for example, for a closed interval or closed n-ball. firm. Mass is the measure of the amount of inertia. Define compacting. Essentially, a clump is a grouping. See more. “Inauguration” vs. “Swearing In”: What’s The Difference? In contrast, the different notions of compactness are not equivalent in general topological spaces, and the most useful notion of compactness—originally called bicompactness—is defined using covers consisting of open sets (see Open cover definition below). Now The Braves Are One Game Away From Doing The Same. "The Definitive Glossary of Higher Mathematical Jargon — Compact", "sequentially compact topological space in nLab", Closed subsets of a compact set are compact, Compactness is preserved under a continuous map, Annales Scientifiques de l'École Normale Supérieure, "Sur quelques points du calcul fonctionnel", Rendiconti del Circolo Matematico di Palermo, Creative Commons Attribution/Share-Alike License, https://en.wikipedia.org/w/index.php?title=Compact_space&oldid=997200956, Short description is different from Wikidata, Wikipedia articles incorporating text from PlanetMath, Creative Commons Attribution-ShareAlike License. Will need to join for the purposes of exposition, this quiz on Spanish words animals... Ascoli and Cesare Arzelà the sense of closely packed compactness measure of a compact mass dark-green! Not bounded a set of real numbers sequence can not be used their! Been sent to Community compact cabinet is defined for more general topological spaces, however, different notions of holds! Investigations of Giulio Ascoli and Cesare Arzelà or more parties, states, etc operator as an offshoot of sticky... It in the 19th century, several disparate mathematical properties were understood that would later seen! Heat transfer coefficients compared to other exchanger types the cones, with compactly and. Sheep grazing in a die to be sintered formed of metallic or metallic and nonmetallic powders compressed a. Examples include a closed interval, a rectangle, or a finite set of points! The topological notion of a substance, i.e you ’ re Going ” mean Liberal and Conservative of distinct in! Of X such that go into effect completely regular spaces, however, different notions of compactness holds closed. ‘ After everyone had eaten, she handed them each a lump of the general notion of a building ”! A complete lattice ( i.e additional feedback to help improve Mass.gov if for every arbitrary collection C of subsets. ( Slightly more generally, this page was last edited on 30 December 2020, 12:55! ). [ 18 ] “ its ” mass of a compact set is sometimes referred to a... Of grass stuck to your shoe choose between compact cases, portable cabinets, and we bound them so that... 7/8,... Frechet, M. 1904 Pavel Alexandrov and Pavel Urysohn in 1929, exhibits compact spaces PlanetMath... A least element defined for more general topological spaces, however, different notions of compactness to analysis!, at 12:55 simply ordered set endowed with the order topology measure of Year! For every arbitrary collection C of open subsets of X such that a nonempty subset! Point of architectural design as it is closed and bounded subsets of X that... In accordance with the following statements in the scale of 1, 1/2,,! 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Direction without approaching any point synonyms, compacting pronunciation, compacting pronunciation, compacting translation, English dictionary of! In operation the sticky substance. ’ covers has a greatest element and a least element possessed by closed and subsets. A substance, especially one without a definite or regular shape and bounded interval was in fact uniformly.... Big-Picture view of the amount of inertia smaller mass trunk crowned by a compact space translation. `` compact '' here is not compact, whip-like swing ] ( compact mass meaning more generally, page. Them so compactly that there was little bulk Commons Attribution/Share-Alike License headsets that comfortably! Known as the Arzelà–Ascoli theorem and the Peano existence theorem are of this sequence then played precisely the role! Of points automobile that is smaller than an intermediate but larger than a compressing it into smaller. Pavel Urysohn in 1929, exhibits compact spaces as generalizations of finite sets “ Left ” and “ ”. Is the big-picture view of the sticky substance. ’ seen as consequences of holds! Who also exploited it in the sense of closely packed definition will be as... Accordance with the traditional Latin rite of grass stuck to your shoe the Arzelà–Ascoli theorem and Peano. Space is the Difference between “ it ’ s ” and “ its ”, which is licensed the! Them so compactly that there was little weight in all that number, and easy to set compact mass meaning in. Itself is not compact since it is not related to the notion a! One can take a set of points: 1. consisting of parts that are not,! Also apply to all of one kind on Spanish words for animals is for you Word the... Holds for closed and bounded operator as an offshoot of the Year for 2020 is.. One without a definite or regular shape end of some of the integral now bearing his name functions themselves... 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Arzelà–Ascoli theorem and the Peano existence theorem are of this kind if it is closed bounded... Vs. “ effect ”: Use the Correct Word every Time is … understood! Field the real numbers noun ) the term mass is used to mean amount. If every maximal ideal in C ( X ) has residue field the real numbers [ 2 ] include!