** Hausdorff maximal principle: In any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset.

In general topological spaces, however, the different notions of compactness are not necessarily equivalent, and the most useful notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, involves the existence of certain finite families of open sets that " cover " the space in the sense that each point of the space must lie in some set contained in the family.

In general topological spaces, however, the different notions of compactness are not equivalent, and the most useful notion of compactness — originally called bicompactness — involves families of open sets that " cover " the space in the sense that each point of the space must lie in some set contained in the family.

Every subset A of the vector space is contained within a smallest convex set ( called the convex hull of A ), namely the intersection of all convex sets containing A.

* The cofinality of a partially ordered set with greatest element is 1 as the set consisting only of the greatest element is cofinal ( and must be contained in every other cofinal subset ).

The halves contained the same set of data and were identical, except the upper half had two holes for a string or twine to go through.

During the 1880s, five foreign editions contained two revelations to John Taylor that were received in 1882 and 1883 ; these revelations " set in order " the priesthood, gave more clarification about the roles of priesthood offices — especially the Seventy — and required Priesthood leaders to live plural marriage in order to qualify to hold their church positions.

Before the entry into force of Protocol 11, Section II ( Article 19 ) set up the Commission and the Court, Sections III ( Articles 20 to 37 ) and IV ( Articles 38 to 59 ) included the high-level machinery for the operation of, respectively, the Commission and the Court, and Section V contained various concluding provisions.

It states that in any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset.

The Hausdorff maximal principle states that, in any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset.

The set contained an audio commentary with director Michael Lehmann, producer Denise Di Novi and writer Daniel Waters, a 30-minute documentary titled Swatch Dogs and Diet Cokeheads, featuring interviews with Ryder, Slater, Doherty, Falk, Lehmann, Waters, Di Novi, Director of Photography Francis Kenny and Editor Norman Hollyn.

There was also a boxed set called variously The Factory Set and One With Everything which contained one of each card from the main set of the limited edition, multiples of the Illuminati cards, and a few cards previously only available in magazines.

The London Regional Transport Act contained provision for setting up subsidiary companies to run the Underground and bus services and in 1985 London Underground Limited ( LUL ), a wholly owned subsidiary of London Regional Transport, was set up to manage the tube network.

A measure can be extended to a complete one by considering the σ-algebra of subsets Y which differ by a negligible set from a measurable set X, that is, such that the symmetric difference of X and Y is contained in a null set.

Then N < sub > x </ sub > is a directed set, where the direction is given by reverse inclusion, so that S ≥ T if and only if S is contained in T. For S in N < sub > x </ sub >, let x < sub > S </ sub > be a point in S. Then ( x < sub > S </ sub >) is a net.

In addition to two or three alliterations, the odd numbered lines had partial rhyme of consonants with dissimilar vowels, not necessarily at the beginning of the word ; the even lines contained internal rhyme in set syllables ( not necessarily at the end of the word ).

He kept out of Epistolae familiares a special set of 19 controversial letters called Liber sine nomine that contained much criticism of the Avignon papacy.

The LSI-11 ( PDP-11 / 03 ), introduced in February, 1975 was the first PDP-11 model produced using large-scale integration ; the entire CPU was contained on four LSI chips made by Western Digital ( the MCP-1600 chip set ; a fifth chip could be added to extend the instruction set, as pictured on the right ).

* In any ring R, a maximal ideal is an ideal M that is maximal in the set of all proper ideals of R, i. e. M is contained in exactly 2 ideals of R, namely M itself and the entire ring R. Every maximal ideal is in fact prime.

Initially set up under the HVP as a rocket training battery ( Number 444 ), Heimat-Artillerie-Park 11 Karlshagen / Pomerania ( HAP 11 ) also contained the A-A Research Command North for the testing of anti-aircraft rockets.

For the Farewell to the World concerts Crombie designed the set, while Mombassa and Seymour designed promotional materials and artwork.

This core memory board, Part Number 50823 D8 7504-14166, and with layout artwork copyrighted 1971 by DGC, was organized in planar fashion as four groups of four banks, each bank carrying two sets of core in a 64 by 64 matrix ; thus there were 64 x 64 = 4096 bits per set, x 2 sets giving 8, 192 bits, x 4 banks giving 32, 768 bits, x 4 goups giving a total of 131, 072 bits, and this divided by the machine word size of 16 bits gave 8, 192 Words of memory.

In early November 2005, Staind released the limited edition 2-CD / DVD set of Chapter V. The set included several rarities and fan favorites — music videos ; a complete, 36-page booklet with exclusive artwork ; an audio disc with an acoustic rendition of " This is Beetle "; the original, melodic rendition of " Reply "; the previously released B-side singles " Novocaine " and " Let It Out "; and live versions of " It's Been Awhile " and " Falling ", among many others.

After rehearsing and writing songs over the summer, they recorded their first demo in late August, making 100 cassettes, with artwork by Squire, and set about trying to get gigs.

Fractals are sometimes combined with human-assisted evolutionary algorithms, either by iteratively choosing good-looking specimens in a set of random variations of a fractal artwork and producing new variations, to avoid dealing cumbersome or unpredictable parameters, or collectively, like in the Electric Sheep project, where people use fractal flames rendered with distributed computing as their screensaver and " rate " the flame they are viewing, influencing the server, which reduces the traits of the undesirables, and increases those of the desirables to produce a computer-generated, community-created piece of art.

Furthermore, the artwork for the song " November has Come " off of the Gorillaz ' 2005 album Demon Days has a picture of a calendar pinned to a brick wall set to the month of October with all the dates marked off in red Xs ( reminiscent of the opening scene in Day of the Dead ).

This event shows recreations of artwork with people posing in a set that looks like the original art.

Although the series is set during the 22nd and 23rd centuries, its retrofuturistic story and artwork are influenced by the post-war culture of 1950s America, and its combination of hope for the promises of technology and lurking fear of nuclear annihilation.

The remasters were made available separately with expanded artwork, and also together in a collector's box set, entitled Boxed.

January 2008 saw the release of ANS7 which expanded the base set to 55 new characters and was the last set to feature new artwork until 2010's Flashback Series 1 subset of six previously unpublished kids.

Each ANS also featured special " chase cards " randomly inserted in packs ; for example, Foil Cards show characters from the original series ( with modified artwork due to the lawsuit ), Scratch ' n Stink cards, Collectible Card Game cards, Temporary Tattoos, 3D Pop-Up cards, Alphabet cards, Activity cards, Magnets, lenticular Loco Motion cards, and die-cut Jigsaw Puzzle cards, along with special bonus cards available only at participating retailers in either " bonus boxes " or rack-packs ; these bonus stickers were the first cards not to have a twin set.

Famed comic-book artist Jack Kirby was even contracted to produce artwork for set design.

These tend to include elaborate stage lighting ; an image magnification ( IMAG ) system and / or pre-recorded video ; inflatables, artwork or other set pieces ; various special effects such as theatrical smoke and fog and pyrotechnics ; and unusual costumes or wardrobe.

The government of Quebec requires that the budget for all new publicly funded buildings set aside 1 % for artwork.

The limited edition box set ( only 1, 000 available ) included remastered versions of the album's track listing, extended artwork, handwritten lyrics, unreleased photos, guitar picks, a commemorative tour laminate, all encased in an intricate Swiss army knife case.

Another popular series was the Civil War News set, also with Norman Saunders ' artwork.

This series contained artwork from previously released cards, the spinoff posters, as well as unreleased artwork from both the original and the aborted 1992 set to make a set of 72 base cards.

The bridge itself rises up from the river, light-flared and enormous, like the outdoor set for an epic opera.

Mother and son recognize each other and, in Mann's version of this legend, make a remarkable confession of guilt to each other, the confession of unconscious motive and unconscious knowledge of their true identities from the time they had first set eyes on each other.

Besides showing no inclination, apparently, to absent himself from his native region even for short periods, and in addition writing a shelf of books set in the region, he has handled in those books an astonishingly complete list of matters which have been important in the South during the past hundred years.

A useful comment on his relation to his region may be made, I think, by noting briefly how in handling Southern materials and Southern problems he has deviated from the pattern set by other Southern authors while remaining faithful to the essential character of the region.

For by now the original cause of the quarrel, Philip's seizure of Gascony, was only one strand in the spider web of French interests that overlay all western Europe and that had been so well and closely spun that the lightest movement could set it trembling from one end to the other.

I don't know how and I don't know why but the two stores, the one in Margaretville and the one in Fleischmanns that had been set up as a partnership, were dissolved, separated from each other.

He could produce carefully constructed orations, set and formal speeches, artfully and prayerfully made by writing and rewriting with all the aid his tutor and others could provide, and then delivered verbatim from memory.

Even to be `` from hope and fear set free '' is at least better than to have lost the first without having got rid of the second.

This central episode consists of a series of staccato scenes set in the period from the beginning of the present century up to the first World War.

The bodice beneath was buttoned and, withdrawing his lips from hers, he set her upright on his knee and started to undo it, unhurriedly as if she were a child.

To be eligible to borrow from a Bank for Cooperatives, a cooperative must be an association in which farmers act together in processing and marketing farm products, purchasing farm supplies, or furnishing farm business services, and must meet the requirements set forth in the Farm Credit Act of 1933, as amended.

In the event the rupees set aside for loans under Section 104 ( E ) of the Act are not advanced within six years from the date of this Agreement because the Export-Import Bank of Washington has not approved loans or because proposed loans have not been mutually agreeable to the Export-Import Bank of Washington and the Department of Economic Affairs of the Government of India, the Government of the United States of America may use the rupees for any purpose authorized by Section 104 of the Act.

Trim excess clay from around lip of mold and set aside while assembling lid.

You then descended one story, glommed a television set from the music room -- the only constructive feature of your visit, by the way -- and, returning to the ground floor, entered the master bedroom.

It appears possible to set a lower limit of about Af for the activation energy of the abstraction of a chlorine atom from a carbon tetrachloride molecule by a chlorine atom to form Af radical.

This method in general solved the problem of obtaining fairly equal concentrations of reactants in each of the six cells from a set.

The binomial probability distribution may describe the variation that occurs from one set of trials of such a binomial experiment to another.

The fact that there can not be any limit points of the set except in closed intervals follows from the argument used in Lemma 1, namely, that near any tangent point in the C-plane the curves C and Af are analytic, and therefore the difference between them must be a monotone function in some neighborhood on either side of the tangent point.

The returns from companies classified as large businesses were set aside and not used because they were not relevant to a study of the opinions and practices of small firms.

The authors set about answering this fundamental question through a detailed investigation of the patient's ability, tactually, ( 1 ) to perceive figure and ( 2 ) to locate objects in space, with his eyes closed ( or turned away from the object concerned ).

These areas, by virtue of their abrupt density of pattern, stated the literal surface with such new and superior force that the resulting contrast drove the simulated printing into a depth from which it could be rescued -- and set to shuttling again -- only by conventional perspective ; ;

Lime white, hard and brilliant, has a tendency to `` jump '' away from the other colors in drying, and also by its capacity to set, to preclude the use of ready-made gradations, so useful in decorative work.

In Pimen's cell the soft prayers of the monks, heard from offstage, not only help to set the scene but emphasize the contrast between young Grigori's thoughts and his situation.

A useful by-product of this system is that the information necessary to set the gyro drift biases is available from the currents necessary to hold the system in level.

Information is also available from this autocollimator system to set the drift bias for the Z-axis gyro.