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# Class Note for MATH 281 at GW

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Date Created: 02/07/15
Math 281 General Topology Notes De nition 1 Relation Let A and B be sets a relation from A a B is a subset R C A X B A relation on A is a relation from A to itself De nition 2 Composition of Relations Given R C A X B S C B X C the composition S o R C A X C de ned as SoR ac3bEB with ab ER and bc ES SoR m U aRb means ab E R De nition 3 Inverse Relation IfR A X B then R 1 B X A R 1 bala7 b E R De nition 4 Identity Relation I C A X A A E A RoIR 038 always Notice that R o R 1 a I in general Example 1 R is the empty relation If R C A X B is a function write R A a B then R 1 a function if and only if R is bijective De nition 5 Equivalence Relation R is an equivalence relation if its reflexive transitive and symmetric De nition 6 Antisymmetric if aRb and bRa then a b De nition 7 Partial Order R is a partial order if it is reflective transitive and antisymmetric De nition 8 Total Order A Partial Order is a total order or a linear order or an order if Va y E A either dRy or ny Munkres gives axioms for lt7 rather than S De nition 9 Quasi Order A relation 3 on A is a quasi order or preorder if it is reflective and transitive Example 2 Let A some collection of sets For ST E A de ne S S T to mean 3 an injective function f S a T This is a quasi order where as containment is a partial order Theorem 1 Schroder Cantor Bernstein HA 3 B and B S A then 3 a bijective from A to B De nition 10 Given a set A a partition 7r ofA is a collection of nonempty pairwise disjoint subsets ofA having union equal to A Note A partition 7r of A is itself a set of sets of A The elements of 7r are called blocks Partition of A gt Equivalence Relation Given a partition 7r 6 HA set of all partitions of 7139 De ne a NW y to mean Ly belong to the same block7 Vw7y E A Then NW is an equivalence relation De nition 11 Equivalence Class Let N be an equivalence relation on A then y E A y N w is called the equivalence class of w Given an equivalence relation N on A7 we obtain a partition A N of A with blocks De nition 12 Quotient Set The partition A N is called the quotient set ofA by N So we have a bijection HA lt gtEquivA7 7r gt gtN7r and A Ngt gtN They have inverse correspondences 7r gt gtN7rgt gt A NW 71 De nition 13 Image ofa Function and Kernel of a Function Given afunction between f A a B we have image imf w E A Q B and the kernel kerf f 1b b E imf Corresponding equivalence relation7 a N y if and only if fw fy Given a group homomorphism7 f G a H7 Kerf f 1e Q G Kerf K is a normal subgroup of G Relationship7 kerf is the partition of G into cosets of K Kerf Suppose f A a B is an abitrary function7 let p be the following map p A a kerf p is called the canonical surjection j imf a B a gt gt w j is called the inclusion mapinjection f kerf a imf a gt gt fV E f is a bijection 7 The canonical factorization of f is f j o f o p The commutative diagram is A L B pl l ker f L im f September 8 2006 De nition 14 Order Preserving If P7Q are partially ordered sets with relations ltp ltQ then a function f P a Q is order preserving if ltp y gt f ltQ y De nition 15 Isomomorphism Let P and Q be partially ordered sets Then a function f P a Q is an isomorphism iff is a bijection and f and f 1 are order preserving De ne a relation on Q by a y ltgt a S yandy S a It is easy to check that this is an equivalence relation and if we de ne a relation 5 on the set of equivalence classes Q by j y ltgt a S y7 then 5 is a well de ned partial order Example 3 Let S all set in some universal set U De ne A S B to mean 3 an injective function f A a B Equivalence classes A are called cardinal numbers or cardinalities We can then restate Schroder Cantor Bernstein as A B ltgt 3 a bijection A lt gt B S is a quasi order 5 is a partial order linear order on the set of all cardinal numbers If P and Q are isomorphic ordered sets7 then all other theoretic properties that hold for P also hold for Q Example 4 Z S Z Z has a minimum element Z doesn t ZS Q ForallaltcEQ73bEQwithaltbltc NottrueinZ What about 1R7 R 9 Yes pZR R gt gtez What about 711701 Yes 7170 01 1 2 gt gt What about 71717R Yes 7171 R x x l gt 717 2 So a7 b R7Va7b 6 IR If PC are posets7 the direct product ordering on P X Q is given by 11751 S 12752 ltgt lt11 S lt12 and b1 S b2 3 17a 17b De nition 16 Dictionary Order If PC are linearly ordered the dictionary order lexicographic order on P X Q is given by a17b1lta27b2ltgt either a1 lt a2 or al a2 and b1 lt b2 De nition 17 Maximal Element Suppose P is a poset and A Q P a E A is maximal ifx S an E A De nition 18 Maximum Element Suppose P is a poset and A Q P a E A is a maximum in A if it is the unique maximum element De nition 19 Bounded Above Suppose P is a poset and A Q P Then A is bounded above if 3p 6 P such that x S p7Vx E A Minimal7 minimum7 and bounded below de ned similarly De nition 20 Least Upper Bound y is a least upper bound for A ify is an upper bound for A and y is the minimal element of the set of all upper bound for A Notation7 least upper bound y lubA supA Greatest lower bound7 in mum7 glbA infA de ned similarly Example 5 A 071 C IR has lubA 1 A x E le lt has no lub in Q n IR lubA Theorem 2 Completeness oflR Every subset oflR which is bounded above has a least upper bound Note The least upper bound property implies the greatest lower bound property 4 September 13 2006 Properties of IR 0 Risa eld o lfxltygtx2ltyzV2 o ifxltygtx2ltyzVzgt0 o completeness every nonempty subset of IR which is bounded above has a least upper bound or supremum onlt2ER73ywithxltyltz De nition 21 Natural Numbers Z lN 1237 The natural numbers has the following properties 0 Principal of Mathematical Induction Suppose A Q lN ifl E A and n E A gt n 1 E A then A lN o Well Ordering Every nonempty subset ole has a least element Note IR7 lt is not well ordered Theorem 3 I ltgt De nition 22 Cartesian Product Suppose A is a collection of sets An indexing function for this family is a surjectiue function f I a A where I is some set called an indexing set We write Ai for fiVi E I Generalizing notation A17A27 An this is the case I 17 27 Note f being surjective means every A E A is equal to Ai for some i E I f is not necessarily injective7 this mean we can have Ai A7 for i a j De ne LJAi7 WAi7 ifI 17 27 H7727 then Uid Ai U221 Ai A1UA2 U UAn write Aihd ieI ieI or AZ 7 for the indexed family De nition 23 I tuple Given an indexed family Aiken an I tuple is afunction x I a UAi such that E A 7Vi E 2 61 I write xi for xi and x id for I tuple x In the case ofI 17237 H7727 then we have x id x17 xn7x E A2171 1 n Cartesian product H Ai consists of all I tuples x ieb xi 6 Ai 2 61 ForI 17237 n7HA HAZ A1 gtlt A2 gtlt gtlt An xL7 6 Ai for i S i S ieI i1 we have Al X A2 X 1727 2 E E ielN i21 i1 HA Av12 ThenHA A X A X A w um i1 n AgtltAgtlt gtltA z Hit it A De nition 24 Finite Set A set A is nite ltgt 3 a bijection A lt gt 127 for some n E N ltgt there is no bijection A lt gt B where B Q A ltgt every injection A a A is bijective ltgt every smjection is bijective De nition 25 A set A is in nite if not nite Example 6 N is in nite f N a 2N n gt gt 2n is bijective De nition 26 Cardinality The cardinality lAl of a set A to be the equivalence class consisting of all sets B such that 3 a bijection B lt gt A Write A B if 3 a bijection A lt gt B ie if lAl Write lAl n if A 177n for some n De nition 27 lAl 3 Bl lAl 3 Bl if 3 an injection from A a Beqnivalently if 3 a snrjection from B a A Theorem 4 S is a partial order on the set of all cardinal numbers Proof re exive lAl S lAl by the identity A a A transitiVity Composition of injections is an injection antisymmetry lAl 3 Bl and lBl S lAl gt lAl lBl is Schoder Cantor Bernstein Uses axiom of choice D The ordering S on cardinals corresponds to the usual ordering for nite cardinals 17 27 We write No for llNl rst in nite cardinal We write N1 for Question ls there a cardinal number 7 With N0 lt 7N1 Continuum Hypothesis There is no such cardinal Theorem 5 Cohen Continuum Hypothesis is neither true of false It is independent of the axioms of set theory De nition 28 Countably In nite A set A is countably in nite if A N0 HN means A is in nite and we can write A 0417127 De nition 29 Countable A set A is countable if it is nite or countably in nite ie if A S No So A is countable ltgt 3 an injection A a N ltgt 3 a surjection N a A Theorem 6 o ifA is countable and B Q A then B is countable o ifA and B are countable then A X B is countable Question If A17 A27 are countable7 is H AZ No 2 1 Proof 1 Given B gt A N D Proof 2 Given A 04170427 Given B b17b27 So7 11751 11752 11753 12751 12752 12753 A X B 13751 13752 13753 So A X B aL7 b17 a27 b17 aL7 b27 a37 b17 a27 b27 aL7 b37 aL7 b47 de nes a bijection A X B a N D

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