Pseudo-intersectionFrom Wikipedia, the free encyclopedia

Contents

1 Finite intersection property 11.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Fraktur 32.1 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Fraktur traditions after 1941 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 Fraktur in Unicode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.6 Typeface samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Innite set 113.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Pseudo-intersection 134.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.2.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

i

Chapter 1

Finite intersection property

In general topology, a branch of mathematics, a collectionA of subsets of a setX is said to have the nite intersectionproperty (FIP) if the intersection over any nite subcollection of A is nonempty. It has the strong nite intersectionproperty (SFIP) if the intersection over any nite subcollection of A is innite.A centered system of sets is a collection of sets with the nite intersection property.

1.1 DenitionLetX be a set withA = fAigi2I a family of subsets ofX . Then the collectionA has the nite intersection property(FIP), if any nite subcollection J I has non-empty intersectionTi2J Ai:1.2 DiscussionClearly the empty set cannot belong to any collection with the nite intersection property. The condition is triviallysatised if the intersection over the entire collection is nonempty (in particular, if the collection itself is empty), andit is also trivially satised if the collection is nested, meaning that the collection is totally ordered by inclusion (equiv-alently, for any nite subcollection, a particular element of the subcollection is contained in all the other elements ofthe subcollection), e.g. the nested sequence of intervals (0, 1/n). These are not the only possibilities however. Forexample, if X = (0, 1) and for each positive integer i, Xi is the set of elements of X having a decimal expansion withdigit 0 in the i'th decimal place, then any nite intersection is nonempty (just take 0 in those nitely many places and1 in the rest), but the intersection of all Xi for i 1 is empty, since no element of (0, 1) has all zero digits.The nite intersection property is useful in formulating an alternative denition of compactness: a space is compact ifand only if every collection of closed sets satisfying the nite intersection property has nonempty intersection itself.[1]This formulation of compactness is used in some proofs of Tychonos theorem and the uncountability of the realnumbers (see next section)

1.3 ApplicationsTheorem. Let X be a non-empty compact Hausdor space that satises the property that no one-point set is open.Then X is uncountable.Proof. We will show that if U X is nonempty and open, and if x is a point of X, then there is a neighbourhoodV U whose closure doesnt contain x (x may or may not be in U). Choose y in U dierent from x (if x is in U,then there must exist such a y for otherwise U would be an open one point set; if x isnt in U, this is possible sinceU is nonempty). Then by the Hausdor condition, choose disjoint neighbourhoodsW and K of x and y respectively.Then K U will be a neighbourhood of y contained in U whose closure doesnt contain x as desired.

Now suppose f : N X is a bijection, and let {xi : i N} denote the image of f. Let X be the rst open set

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2 CHAPTER 1. FINITE INTERSECTION PROPERTY

and choose a neighbourhood U1 X whose closure doesnt contain x1. Secondly, choose a neighbourhood U2 U1 whose closure doesnt contain x2. Continue this process whereby choosing a neighbourhood Un Un whoseclosure doesnt contain xn. Then the collection {Ui : i N} satises the nite intersection property and hence theintersection of their closures is nonempty (by the compactness of X). Therefore there is a point x in this intersection.No xi can belong to this intersection because xi doesnt belong to the closure of Ui. This means that x is not equal toxi for all i and f is not surjective; a contradiction. Therefore, X is uncountable.All the conditions in the statement of the theorem are necessary:1. We cannot eliminate the Hausdor condition; a countable set with the indiscrete topology is compact, has morethan one point, and satises the property that no one point sets are open, but is not uncountable.2. We cannot eliminate the compactness condition as the set of all rational numbers shows.3. We cannot eliminate the condition that one point sets cannot be open as a nite space given the discrete topologyshows.Corollary. Every closed interval [a, b] with a < b is uncountable. Therefore, R is uncountable.Corollary. Every perfect, locally compact Hausdor space is uncountable.Proof. Let X be a perfect, compact, Hausdor space, then the theorem immediately implies that X is uncountable.If X is a perfect, locally compact Hausdor space which is not compact, then the one-point compactication of X isa perfect, compact Hausdor space. Therefore the one point compactication of X is uncountable. Since removinga point from an uncountable set still leaves an uncountable set, X is uncountable as well.

1.4 ExamplesA lter has the nite intersection property by denition.

1.5 TheoremsLet X be nonempty, F 2X, F having the nite intersection property. Then there exists an F ultralter (in 2X) suchthat F F.See details and proof in Csirmaz & Hajnal (1994).[2] This result is known as ultralter lemma.

1.6 VariantsA family of sets A has the strong nite intersection property (sp), if every nite subfamily of A has inniteintersection.

1.7 References[1] A space is compact i any family of closed sets having p has non-empty intersection at PlanetMath.org.

[2] Csirmaz, Lszl; Hajnal, Andrs (1994), Matematikai logika (IN HUNGARIAN), Budapest: Etvs Lornd University.

Finite intersection property at PlanetMath.org.

Chapter 2

Fraktur

This article is about the script. For Fraktur folk art, see Fraktur (Pennsylvania German folk art).Fraktur (German: [faktu]) is a calligraphic hand of the Latin alphabet and any of several blackletter typefaces

A modern sans-serif and four blackletter typefaces (left to right): Textur(a), Rotunda, Schwabacher and Fraktur.

derived from this hand. The blackletter lines are broken up that is, their forms contain many angles when com-

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4 CHAPTER 2. FRAKTUR

pared to the smooth curves of the Antiqua (common) typefaces modeled after antique Roman square capitals andCarolingian minuscule. From this, Fraktur is sometimes contrasted with the Latin alphabet in northern Europeantexts, being sometimes called the German alphabet, despite simply being a typeface of the Latin alphabet. Simi-larly, the term Fraktur or Gothic is sometimes applied to all of the blackletter typefaces (known in German asGebrochene Schrift (Broken Script)).Here is the entire alphabet of English in Fraktur, using the \mathfrak font of the mathematical typesetting packageTeX:A B C D E F G H I J K LMNO P Q R S T U VW X Y Z

a b c d e f g h i j k l m n o p q r s t u v w x y z

The word derives from the past participle fractus (broken) of Latin frangere (to break); the same root as theEnglish word fracture.

2.1 CharacteristicsBesides the 26 letters of the Latin alphabet, Fraktur includes the (Eszett [stst]), vowels with umlauts, and the (long s). Some Fraktur typefaces also include a variant form of the letter r known as the r rotunda, and many a varietyof ligatures which are left over from cursive handwriting and have rules for their use. Most older Fraktur typefacesmake no distinction between the majuscules I and J (where the common shape is more suggestive of a J), eventhough the minuscules i and j are dierentiated.One dierence between the Fraktur and other blackletter scripts is that in the lower case o, the left part of the bowis broken, but the right part is not. In Danish texts composed in Fraktur, the letter was already preferred to theGerman and Swedish in the 16th century.[1]

2.2 OriginThe rst Fraktur typeface was designed when Holy Roman EmperorMaximilian I (c. 14931519) established a seriesof books and had a new typeface created specically for this purpose, designed by Hieronymus Andreae. Frakturquickly overtook the earlier Schwabacher and Textualis typefaces in popularity, and a wide variety of Fraktur fontswere carved. It became common in the German-speaking world and areas under German inuence (Scandinavia,the Baltic states, Central Europe). Over the succeeding centuries, most Central Europeans switched to Antiqua, butGerman-speakers remained a notable holdout.

2.3 UseTypesetting in Fraktur was still very common in the early 20th century in all German-speaking countries and ar-eas, as well as in Norway, Estonia, and Latvia, and was still used to a very small extent in Sweden, Finland andDenmark,[2] while other countries typeset in Antiqua in the early 20th century. Some books at that time used relatedblackletter fonts such as Schwabacher; however, the predominant typeface was the Normalfraktur, which came inslight variations.From the late 18th century to the late 19th century, Fraktur was progressively replaced by Antiqua as a symbol ofthe classicist age and emerging cosmopolitanism in most of the countries in Europe that had previously used Fraktur.This move was hotly debated in Germany, where it was known as the AntiquaFraktur dispute. The shift aectedmostly scientic writing in Germany, whereas most belletristic literature and newspapers continued to be printed inbroken fonts. This radically changed on January 3, 1941, when Martin Bormann issued a circular to all public oceswhich declared Fraktur (and its corollary, the Stterlin-based handwriting) to be Judenlettern (Jewish letters) andprohibited their further use.[5] German historian Albert Kapr has speculated that the rgime had realized that Frakturwould inhibit communication in the territories occupied during World War II.[6] Fraktur saw a brief resurgence afterthe War, but quickly disappeared in a Germany keen on modernising its appearance.Fraktur is today used mostly for decorative typesetting: for example, a number of traditional German newspaperssuch as the Frankfurter Allgemeine, as well as the Norwegian Aftenposten, still print their name in Fraktur on themasthead (as indeed do some newspapers in other European countries) and it is also popular for pub signs and the

2.3. USE 5

ACzech example of Fraktur: Title page of eskmarinskmuzika by AdamVclavMichna z Otradovic (1647) (Ceskmarynskamuzyka by old orthography)

like. In this modern decorative use, the traditional rules about the use of long s and short s and of ligatures are oftendisregarded.Individual Fraktur letters are sometimes used in mathematics, which often denotes associated or parallel concepts bythe same letter in dierent fonts. For example, a Lie group is often denoted by G, while its associated Lie algebra isg . A ring ideal might be denoted by a while an element is a 2 a . The Fraktur c is also used to denote the cardinalityof the continuum of the real line. In model theory, A is used to denote an arbitrary model, with A as its universe.

6 CHAPTER 2. FRAKTUR

2.4 Fraktur traditions after 1941Even after the abolition of Fraktur, some editions include elements of it in headlines.Very occasionally, academic works still used Fraktur in the text itself. Notably, Joachim Jeremias work Die Briefean Timotheus und Titus (The Letters of Timothy and Titus) was published in 1963 using Fraktur. More often, someligatures ch, ck from Fraktur were used in antiqua-typed editions.That continued mostly up to the oset type period.

2.5 Fraktur in UnicodeIn Unicode, Fraktur is treated as a font of the Latin alphabet, and is not encoded separately. The additional ligaturesthat are required for Fraktur fonts will not be encoded in Unicode.[7] Instead, Unicode proposes to deal with theseligatures using smart-font technologies such as OpenType, AAT or Graphite. There are many Fraktur fonts that donot use smart-font technologies, but use their own legacy encoding instead that is not compliant with Unicode.There are Fraktur symbols in the Unicode blocks of mathematical alphanumeric symbols and letterlike symbols.However, these are meant to be used only in mathematics.[8] Therefore, letters such as , , , and , which are notused in mathematics, are excluded.

2.6 Typeface samples

In the gures below, the German sentence that appears after the names of the fonts (Walbaum-Fraktur in Fig. 1and Humboldtfraktur in Fig. 2) reads, Victor jagt zwlf Boxkmpfer quer ber den Sylter Deich. It means Victorchases twelve boxers across the Sylt dike and contains all 26 letters of the alphabet plus the umlauted glyphs used inGerman, making it an example of a pangram. Note that in the second specimen, the rst t in Humboldtfraktur isomitted.

2.7 See also

2.8 References[1] Cf., inter alia, Bibla: Det er den gantske Hellige Scit: udst paa Danske. 1550. (Danish) & Biblia: Det er Den gantske

Hellige Scrit paa Danske igien ouerseet oc prentet eter vor allernaadigste herris oc Kongis K. Christian den IV. Befaling.1633. (Danish)

[2] In Denmark in 1902 the percentage of printed material using antiqua amounted to 95% according to R. Paulli, Densejrende antikva, i: Det trykte Ord, published by Grask Cirkel, Copenhagen, 1940.

[3] R. Paulli, Den sejrende antikva, i: Det trykte Ord, published by Grask Cirkel, Copenhagen, 1940.

[4] Rem, Tore (2009). Materielle variasjoner. Overgang fra fraktur til antikva i Norge. In Malm, Mats; Sjnell, BarbroSthle; Sderlund, Petra. Bokens materialitet: Bokhistoria och bibliogra. Stockholm: Svenska Vitterhetssamfundet. ISBN978-91-7230-149-8.

[5] Facsimile of Bormanns Memorandum (in German)The memorandum itself is typed in Antiqua, but the NSDAP letterhead is printed in Fraktur.For general attention, on behalf of the Fhrer, I make the following announcement:It is wrong to regard or to describe the so-called Gothic script as a German script. In reality, the so-called Gothic scriptconsists of Schwabach Jew letters. Just as they later took control of the newspapers, upon the introduction of printingthe Jews residing in Germany took control of the printing presses and thus in Germany the Schwabach Jew letters wereforcefully introduced.

2.9. FURTHER READING 7

Today the Fhrer, talking with Herr Reichsleiter Amann and Herr Book Publisher Adolf Mller, has decided that in thefuture the Antiqua script is to be described as normal script. All printed materials are to be gradually converted to thisnormal script. As soon as is feasible in terms of textbooks, only the normal script will be taught in village and state schools.The use of the Schwabach Jew letters by ocials will in future cease; appointment certications for functionaries, streetsigns, and so forth will in future be produced only in normal script.On behalf of the Fhrer, Herr Reichsleiter Amann will in future convert those newspapers and periodicals that already haveforeign distribution, or whose foreign distribution is desired, to normal script.

[6] Kapr, Albert (1993). Fraktur: Form und Geschichte der gebrochenen Schriften. Mainz: H. Schmidt. p. 81. ISBN 3-87439-260-0.

[7] http://www.unicode.org/faq/ligature_digraph.html#7

[8] Unicode Consortium. Ligatures, Digraphs, Presentation Forms vs. Plain Text.

2.9 Further reading Bain, Peter and Paul Shaw. Blackletter: Type and National Identity. Princeton Architectural Press: 1998.

ISBN 1-56898-125-2.

Silvia Hartmann: Fraktur oder Antiqua. Der Schriftstreit von 1881 bis 1941, Peter Lang, Frankfurt am Mainu. a. 1998 (2. b. A. 1999), ISBN 978-3-631-35090-4

Fiedl, Frederich, Nicholas Ott and Bernard Stein. Typography: An Encyclopedic Survey of Type Design andTechniques Through History. Black Dog & Leventhal: 1998. ISBN 1-57912-023-7.

Macmillan, Neil. An AZ of Type Designers. Yale University Press: 2006. ISBN 0-300-11151-7.

2.10 External links A complete Fraktur chart (German) Website of Dieter Stemann, which has a large number of digitized Fraktur fonts Blackletter: Type and National Identity (German) Delbanco: German Purveyors of Fraktur fonts (commercial) Setting up Microsoft Windows NT, 2000 or Windows XP to support Unicode supplementary characters UniFraktur: Free Unicode-compliant Fraktur fonts and resources

8 CHAPTER 2. FRAKTUR

Front page of Gustav Vasa's Bible from 1541, using Fraktur. The title translated to English reads: The Bible / That is / All the HolyScriptures / in Swedish. Printed in Uppsala. 1541. (Note that Swedish had th like English at the time; it would later change tod.)

2.10. EXTERNAL LINKS 9

Usage map: A map presenting the contemporary German view of the extent of scripts around 1900. In reality only German-speakingcountries, Estonia and Latvia still used Fraktur as the majority script at this time. Denmark had shifted to antiqua during the mid19th century,[3] and in Norway the majority of printed texts used antiqua around 1900.[4]

Fig. 1. Walbaum-Fraktur (1800)

10 CHAPTER 2. FRAKTUR

Fig. 2. Humboldtfraktur (Hiero Rhode, 1938)

Chapter 3

Innite set

In set theory, an innite set is a set that is not a nite set. Innite sets may be countable or uncountable. Someexamples are:

the set of all integers, {..., 1, 0, 1, 2, ...}, is a countably innite set; and the set of all real numbers is an uncountably innite set.

3.1 PropertiesThe set of natural numbers (whose existence is postulated by the axiom of innity) is innite. It is the only set thatis directly required by the axioms to be innite. The existence of any other innite set can be proved in ZermeloFraenkel set theory (ZFC) only by showing that it follows from the existence of the natural numbers.A set is innite if and only if for every natural number the set has a subset whose cardinality is that natural number.If the axiom of choice holds, then a set is innite if and only if it includes a countable innite subset.If a set of sets is innite or contains an innite element, then its union is innite. The powerset of an innite set isinnite. Any superset of an innite set is innite. If an innite set is partitioned into nitely many subsets, then atleast one of themmust be innite. Any set which can be mapped onto an innite set is innite. The Cartesian productof an innite set and a nonempty set is innite. The Cartesian product of an innite number of sets each containingat least two elements is either empty or innite; if the axiom of choice holds, then it is innite.If an innite set is a well-ordered set, then it must have a nonempty subset that has no greatest element.In ZF, a set is innite if and only if the powerset of its powerset is a Dedekind-innite set, having a proper subsetequinumerous to itself.[1] If the axiom of choice is also true, innite sets are precisely the Dedekind-innite sets.If an innite set is a well-orderable set, then it has many well-orderings which are non-isomorphic.

3.2 See also Aleph number Dedekind-innite set Innity

3.3 References[1] Boolos, George (1994), The advantages of honest toil over theft, Mathematics and mind (Amherst, MA, 1991), Logic

Comput. Philos., Oxford Univ. Press, New York, pp. 2744, MR 1373892. See in particular pp. 3233.

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12 CHAPTER 3. INFINITE SET

3.4 External links Weisstein, Eric W., Innite Set, MathWorld.

Chapter 4

Pseudo-intersection

In mathematical set theory, a pseudo-intersection of a family of sets is an innite set S such that each element ofthe family contains all but a nite number of elements of S. The pseudo-intersection number, sometimes denotedby the fraktur letter , is the smallest size of a family of innite subsets of the natural numbers that has the strongnite intersection property but has no pseudo-intersection.

4.1 References Kunen, Kenneth (2011), Set theory, Studies in Logic 34, London: College Publications, ISBN 978-1-84890-

050-9, MR 2905394, Zbl 1262.03001

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14 CHAPTER 4. PSEUDO-INTERSECTION

4.2 Text and image sources, contributors, and licenses4.2.1 Text

Finite intersection property Source: https://en.wikipedia.org/wiki/Finite_intersection_property?oldid=669469162Contributors: MichaelHardy, Revolver, Charles Matthews, Dcoetzee, Ruakh, EmilJ, ABCD, Linas, Yuval Madar~enwiki, R.e.b., YurikBot, Ondenc, Kompik,SmackBot, Mhss, Nbarth, Physis, Zero sharp, Vaughan Pratt, Carl Turner, David Eppstein, Alighat~enwiki, JackSchmidt, J.Gowers,Addbot, Topology Expert, Ozob, Yobot, FrescoBot, Xnn, WikitanvirBot, Mark viking and Anonymous: 9

Fraktur Source: https://en.wikipedia.org/wiki/Fraktur?oldid=656930162 Contributors: The Cunctator, DanKeshet, Montrealais, Some-one else, Shaydon, Spi~enwiki, Bdesham, Patrick, DopeshJustin, Dominus, Menchi, Wwwwolf, Ihcoyc, WeiNix, Djmutex, Error,Nertzy, Adam Bishop, Robbot, Dittaeva, Lowellian, Mirv, Bkell, Kzhr, Jor, Wwoods, Hans-Friedrich Tamke, Jason Quinn, Proslaes,Coldacid, J. 'mach' wust, Beland, Evertype, Lesgles, Jossi, Mzajac, OwenBlacker, LHOON, Chmod007, Poccil, Lovelac7, Hydrox,Dbachmann, Bender235, Cathack, Kwamikagami, Neg, Jumbuck, Eric Kvaalen, DavidHoag, Zippanova, Stephan Leeds, Cmapm, Cy-cler~enwiki, Walshga, JALockhart, Angr, Sburke, Ruud Koot, Nema Fakei, Teemu Leisti, Jorunn, FlaBot, WhyBeNormal, YurikBot,Hydrargyrum, Cryptic, Dtrebbien, Scs, Wikilackey, Alarob, Thnidu, Wikipeditor, Wwzeitler, GrinBot~enwiki, PKtm, Eskimbot, Beta-command, Bluebot, Hongooi, Alphathon, Jukrat, Lambiam, RandomCritic, MottyGlix, Mikelima~enwiki, Charvex, WeggeBot, Icarusof old, AndrewHowse, Aodhdubh, Thijs!bot, Wikid77, Olahus, SeNeKa, Magioladitis, GearedBull, Nyttend, Nikevich, Objectivesea,Nono64, Adavidb, Johnbod, Signalhead, TreasuryTag, JhsBot, This, that and the other, M.thoriyan, AlanUS, Wahrmund, Denisarona,Saddhiyama, Uncle Milty, Tlustulimu, El bot de la dieta, Dthomsen8, Addbot, Conrad Hughes, Ehrenkater, Lightbot, Ajstern, Yobot,Yngvadottir, AnomieBOT, LlywelynII, Algorithme, Xqbot, Omnipaedista, Jangirke, Brian747180, Lotje, JasonSaulG, ZroBot, Water-Crane, DASHBotAV, Bubble3d, ClueBot NG, BG19bot, MiTheFox, Teika kazura, Gregory david baker, Khazar2, AthanasiusOfAlex,Tony Mach, Dahlia.V.K., ReconditeRodent, Zumoarirodoka, Mgkrupa, BronyLyncher and Anonymous: 78

Innite set Source: https://en.wikipedia.org/wiki/Infinite_set?oldid=659075298Contributors: TheAnome, Toby Bartels, DennisDaniels,Charles Matthews, David Shay, Bkell, Tobias Bergemann, Giftlite, Paul August, Rgdboer, Benji22210, Zerofoks, Salix alba, FlaBot,VKokielov, Chobot, DVdm, 4C~enwiki, Grubber, Trovatore, Maksim-e~enwiki, Addshore, Bidabadi~enwiki, Vina-iwbot~enwiki, Lam-biam, StevenPatrickFlynn, Bjankuloski06en~enwiki, Fell Collar, JRSpriggs, CRGreathouse, CBM, Escarbot, JAnDbot, Olaf, David Epp-stein, Ttwo, Maurice Carbonaro, Doug, DFRussia, Cli, JP.Martin-Flatin, Alexbot, Hans Adler, Hatso, Addbot, Favonian, Luckas-bot, TaBOT-zerem, AnomieBOT, JackieBot, Materialscientist, VladimirReshetnikov, Erik9bot, Nicolas Perrault III, BenzolBot, Tku-vho, Pinethicket, SkyMachine, TobeBot, Beyond My Ken, Wgunther, Tommy2010, ZroBot, Donner60, ClueBot NG, Widr, Juro2351,Magic6ball, Sauood07, YFdyh-bot, Blackbombchu, K9re11, Centralpanic and Anonymous: 37

Pseudo-intersection Source: https://en.wikipedia.org/wiki/Pseudo-intersection?oldid=648937330 Contributors: R.e.b. and K9re11

4.2.2 Images File:CardContin.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/75/CardContin.svg License: Public domain Contrib-

utors: en:Image:CardContin.png Original artist: en:User:Trovatore, recreated by User:Stannered File:Commons-logo.svg Source: https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg License: ? Contributors: ? Origi-

nal artist: ? File:Folder_Hexagonal_Icon.svg Source: https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg License: Cc-

by-sa-3.0 Contributors: ? Original artist: ? File:Fraktur_humboldtfraktur.png Source: https://upload.wikimedia.org/wikipedia/commons/4/40/Fraktur_humboldtfraktur.pngLi-

cense: CC-BY-SA-3.0 Contributors: ? Original artist: ? File:Fraktur_letter_A-umlaut.png Source: https://upload.wikimedia.org/wikipedia/commons/c/c9/Fraktur_letter_A-umlaut.png Li-

cense: CC-BY-SA-3.0 Contributors: ? Original artist: ? File:Fraktur_letter_A.png Source: https://upload.wikimedia.org/wikipedia/commons/4/4c/Fraktur_letter_A.pngLicense: CC-BY-SA-

3.0 Contributors: ? Original artist: ? File:Fraktur_letter_B.png Source: https://upload.wikimedia.org/wikipedia/commons/7/72/Fraktur_letter_B.png License: CC-BY-SA-

3.0 Contributors: ? Original artist: ? File:Fraktur_letter_C.png Source: https://upload.wikimedia.org/wikipedia/commons/7/72/Fraktur_letter_C.pngLicense: CC-BY-SA-

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4.2. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 15

File:Fraktur_letter_K.png Source: https://upload.wikimedia.org/wikipedia/commons/a/ab/Fraktur_letter_K.pngLicense: CC-BY-SA-3.0 Contributors: ? Original artist: ?

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4.2.3 Content license Creative Commons Attribution-Share Alike 3.0

Finite intersection propertyDefinitionDiscussionApplicationsExamplesTheorems VariantsReferences

FrakturCharacteristicsOriginUseFraktur traditions after 1941Fraktur in UnicodeTypeface samplesSee alsoReferencesFurther readingExternal links

Infinite setPropertiesSee alsoReferencesExternal links

Pseudo-intersectionReferencesText and image sources, contributors, and licensesTextImagesContent license