A nonempty compact subset of the real numbers has a greatest element and a least element. That is, if Let X be a simply ordered set endowed with the order topology. 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. {\displaystyle K\subset Z\subset Y} ‘After everyone had eaten, she handed them each a lump of the sticky substance.’. In spaces that are compact in this sense, it is often possible to patch together information that holds locally—that is, in a neighborhood of each point—into corresponding statements that hold throughout the space, and many theorems are of this character. Fruit should be firm and excellent in condition. Explore 'compact' in the dictionary. Various equivalent notions of compactness, such as sequential compactness and limit point compactness, can be developed in general metric spaces.[4]. Euclidean space itself is not compact since it is not bounded. [1][2] 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. Freddie Freeman Took The Leap. to crush into compact form for convenient disposal or for storage until disposal: a small case containing a mirror, face powder, a puff, and sometimes rouge. X It was this notion of compactness that became the dominant one, because it was not only a stronger property, but it could be formulated in a more general setting with a minimum of additional technical machinery, as it relied only on the structure of the open sets in a space. firm. Applications of compactness to classical analysis, such as the Arzelà–Ascoli theorem and the Peano existence theorem are of this kind. compacting synonyms, compacting pronunciation, compacting translation, English dictionary definition of compacting. The term mass is used to mean the amount of matter contained in an object. 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. At the end of some of the branches come the cones, with compactly arranged and simple sporophylls all of one kind. This notion is defined for more general topological spaces than Euclidean space in various ways. 2 circumlocutory, garrulous, lengthy, long-winded, prolix, rambling, verbose, wordy. • COMPACT (adjective) The Most Surprisingly Serendipitous Words Of The Day. American Public Human Services Association 1133 Nineteenth Street, NW Suite 400 Washington, DC 20036 (202) 682-0100 fax: (202) 289-6555 R A space X is compact if its hyperreal extension *X (constructed, for example, by the ultrapower construction) has the property that every point of *X is infinitely close to some point of X⊂*X. an automobile that is smaller than an intermediate but larger than a. 1 (adjective) in the sense of closely packed. the compact body of a lightweight wrestler. English Collins Dictionary - English synonyms & Thesaurus. Y If you haven’t heard of the multi-state nursing license compact, it’s time to find out how this great program can streamline your eligibility for a variety of travel nursing opportunities—and how some recent changes might affect you. 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). all subsets have suprema and infima).[18]. 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) 1 A compact mass of a substance, especially one without a definite or regular shape. (Slightly more generally, this is true for an upper semicontinuous function.) We need some definitions first. US Federal Government Executed 13 Inmates under Trump Administration 1/18/2021 - On Jan. 16, 2021, the federal government executed Dustin Higgs, the thirteenth and final prisoner executed under the Trump administration, which carried out the first federal executions since 2003. For other uses, see, Topological notions of all points being "close". In entomology, specifically, compacted or pressed close, as a jointed organ, or any part of it, when the joints are very closely united, forming a continuous mass: as, a compact antennal club; compact palpi. K The term compact set is sometimes used as a synonym for compact space, but often refers to a compact subspace of a topological space as well. Compaction definition is - the act or process of compacting : the state of being compacted. 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. These are compact, over-ear headsets that rest comfortably, and that comfort is helped by the lightweight materials used in their construction. A topological space X is pseudocompact if and only if every maximal ideal in C(X) has residue field the real numbers. Explore 'compact' in the dictionary. This property was significant because it allowed for the passage from local information about a set (such as the continuity of a function) to global information about the set (such as the uniform continuity of a function). 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. What Is The Difference Between “It’s” And “Its”? 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). A compact is a signed written agreement that binds you to do what you've promised. 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). How to use mass in a sentence. As a Euclidean space is a metric space, the conditions in the next subsection also apply to all of its subsets. 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. 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. Synonyms. Essentially, a clump is a grouping. In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness. Originally developed in 2000, by … That is, K is compact if for every arbitrary collection C of open subsets of X such that. 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. 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). It was Maurice Fréchet who, in 1906, had distilled the essence of the Bolzano–Weierstrass property and coined the term compactness to refer to this general phenomenon (he used the term already in his 1904 paper[7] which led to the famous 1906 thesis). Compactness is a "topological" property. It also refers to something small or closely grouped together, like the row of compact … [13] There are pseudocompact spaces that are not compact, though. 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. The Heine–Borel theorem, as the result is now known, is another special property possessed by closed and bounded sets of real numbers. See more. Examples include a closed interval, a rectangle, or a finite set of points. 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. 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. “Affect” vs. “Effect”: Use The Correct Word Every Time. We would also like a characterization of compact sets based entirely on open sets. 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.

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