# pseudo intersection

Post on 16-Dec-2015

220 views

Category:

## Documents

Embed Size (px)

DESCRIPTION

TRANSCRIPT

• 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

1

• 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

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-

3

• 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 r