الجـامعــــــــــة اﻹســـــﻼميــة بغــزة The Islamic University of Gaza عمادة البحث العلمي والدراسات العليا Deanship of Research and Graduate Studies كـليــــــــــــــــــــة العـلـــــــــــــــــــــوم Faculty of Science Master of Mathematical Science ماجستيـــــــر العلـــــــوم الـريـاضـيــة Topological Properties of the Cantor Spaces الخصائص التبولوجية لفضاءات كانتور By Ahamed M. Alnajjar Supervised by Prof. Dr. Hisham B. Mahdi Professor of Mathematices A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematical Sciences. October/2018 إقــــــــــــــرار أنا الموقع أدناه مقدم الرسالة التي تحمل العنوان: Topological Properties of the Cantor Spaces ﺍلﺨﺼائص ﺍلتبولوﺟﻴة لﻔضاءات ﻛانتوﺭ أقر بأن ما اشتملت عليو ىذه الرسالة إنما ىه نتاج جيدي الخاص، باستثناء ما تمت اﻹشارة إليو حيثما ورد، وأن ىذه الرسالة ككل أو أي جزء منيا لم يقدم من قبل اﻻخرين لنيل درجة أو لقب علمي أو بحثي لدى أي مؤسدة تعليمية أو بحثية أخرى. Declaration I understand the nature of plagiarism, and I am aware of the University’s policy on this. The work provided in this thesis, unless otherwise referenced, is the researcher's own work, and has not been submitted by others elsewhere for any other degree or qualification. اسم الطالب: أحمد دمحم خالد النجار Student's name: Ahamed M. Alnajjar التهقيع: :Signature التاريخ: :Date i Abstract ﻓﻲ هذه الرسالة ، نتحدث بشكل رئيسﻲ عن ﻓضاءات كانتور ومكعب كانتور ﻷية مجموعات غير منتهيه ﻓﻲ X ﻓان مكعب كانتور تم تعريفه على انه مجموعات الطاقة من (P(X من X. وهذا الفضاء له تمثيﻼت أخرى ويتكون اﻷساس لهذا الفضاء من كل الفترات التﻲ تكون على شكل [ك, م] حيث ان ك منتهيه و م غير منتهيه وحيث تم استخدام هذا اﻷساس بفعاليه ودرسنا خصائص الفضاءات الكانتور و مكعب كانتور وأنواع مختلفه من مكعب كانتور ,وأثبتنا أن الفضاء كانتور هو الفضاء الوحيد (ما يصل إلى التماثل) التﻲ تجمع بين خصائص الفضاء المنفصل ، والكمال ، والمدمج ، والمتر المترابط تما ًما وأثبتنا أن µ تكون تبولوجﻲ ألكساندر وف ﻓإنها تكون مغلقه ﻓﻲ 푿 . وعﻼوة على ذلك ﻓإن أصغر تبولوجﻲ ألكساندر وف هﻲµ Abstract In this thesis, we mainly talk about the Cantor space and the Cantor cube. For any infinite set X, the Cantor cube is defined as the power set P(X) of X with the product topology. This space has other representation 2X . The base for this space consists of all intervals of the form [K; F ] where K is finite set and F is cofinite set, and this base has been used effectively. We study the properties of the Cantor space, the Cantor cube 2X and the properties of different types of cubes. We prove that the Cantor space is the only space (up to homeomorphism) that combines the properties of totally disconnected, perfect, compact, and metric space. We prove that τ on X is Alexandroff topology iff τ is closed in 2X . Moreover, if τ is topology on X, then the closure τ of τ in 2X is the smallest Alexandroff topology in X containing τ . iii Acknowledgements After God, I appreciate the greatest appreciation of my parents who continued my support and pray for me to continue until the end. I thank my wife and all my brothers, friends and loved ones for sharing my scientific joy. I owe them my deepest love. My thanks go to professor Hisham Mahdi who supported me and supported me to continue this thesis. And always gave me suggestions and guidance, and I would like to thank all the faculty in the Department of Mathematics, especially professor Asaad Asaad, professors Ayman al-Sakka, Dr. Ahmed Mabhouh and profesor, God's mercy upon him, Eissa al-Habil. iv Contents Abstract iii Acknowledgements iv Contentsv Introduction1 1 Preliminaries5 1.1 Properties of topological spaces....................5 1.2 Directed sets and nets......................... 10 1.3 Product topology............................ 12 2 Alexandroff Spaces 15 2.1 Partially ordered sets.......................... 15 2.2 Alexandroff topology and minimal open neighborhoods....... 18 2.3 T0-Alexandroff space viewed as poset................. 21 2.4 Product of Alexandroff space..................... 24 v 3 Cantor Space 27 3.1 Construction of the Cantor set..................... 28 3.2 Equivalence representation of the Cantor space............ 29 3.2.1 The sequence representation.................. 29 3.2.2 The representation of C as power set of N .......... 31 3.3 Properties of the Cantor space..................... 31 4 The Cantor Cube of Infinite set X 39 4.1 The Cantor cube 2X .......................... 39 4.2 Properties of different types of cube.................. 42 4.3 More on the Cantor cube 2X ..................... 44 Bibliography 53 vi Introduction The Cantor set is a subset of the interval [0,1] and has many definitions and many different constructions. The Cantor set is a set of points in the line. It was defined by the German mathematician Georg Cantor in 1874 and introduced in 1883. The Cantor set has laid a modern foundation in the category of the topology. Although Cantor originally provided a purely abstract definition, the most accessible is the middle-thirds or ternary set construction. For the Cantor space, we attempt to a systematic study of analytic topologies over the natural numbers N (or any count- able set X). We identify every subset of N with its characteristic function. This implies that the power set P(N) is identified as the Cantor space 2N. Now and for more general, Let X be an infinite set, P(X), the power set of X is defined as the product space 2X = f0; 1gX , the collection of all function from X to f0; 1g where f0; 1g has discrete topology. In this case, the space 2X is called a cantor cube of X. The idea of this thesis is how to systematically study the types of topology within the Cantor cube. The thesis depends basically on the results of the paper ”Alexandroff topologies viewed as closed sets in a Cantor cube". The goal is to develop this research on other characteristics of the topology. Alexandroff spaces were first introduced by P. Alexandroff in 1937 under the name of Diskrete Raume 1 (discrete space). An Alexandroff topology is a topology has the additional property arbitrary intersection of open sets is open. In this paper we present some results of Alexandroff topologies. This thesis consists of four chapters. Chapter one contains three sections about preliminaries that will be used in the remainder of this the- sis. In section one, we talk about the basic definitions and theorems of topological spaces. In the second section, we give an introduction to directed sets and nets. We study some important theories that will be used in this thesis. The last section take about the product topology. Chapter two studies the Alexandroff topology and its topological properties. This chapter has been divided into four sections. The first section studies partially order sets. In the second section, we study the Alexandroff topology and minimal open neighborhoods. It is easy to verify that a T topology is Alexandroff Topology iff for every x 2 X, the set Nx = fU : x 2 V and V 2 τg is an open set. Nx is called the irreducible (or minimal) neighborhood of x and one can easy check that a collection A of subsets of X has minimal set T Bx of x 2 X in A iff fV 2 A : x 2 Ug 2 A. In third section we prove that each T0-Alexandroff space has a poset in 1 − 1 and onto way such that each one is completely determined by the other. In last section, we study the product of Alexandroff topology. Chapter three is talking about the Cantor set. In the first section, we study the construction of Cantor set. And we will show that it is non empty set. Thus, the Cantor set is a subset of I formed from deleting accountable union of open interval. If we consider C as a subspace of I with its usual topology, C is called the Cantor space. We will study the properties of this space in the next section. But in this section, we give some representation of a space equivalent 2 (up to homeomorphic) to the Cantor space. From now an, we will use any rep- resentation of them as the Cantor space. In second section, we study equivalence representation of the Cantor space. There are many equivalence representations but we will give only two of them. One representation is the sequence represen- tation and second one is the representation of C as a power set of N. In the last section, we study the properties of the Cantor space. The Cantor space contains some important properties. The Cantor set is a closed subset in [0; 1], totally dis- connected and a perfect space. Chapter four consists of three sections. The first one is talking about the Cantor cube 2X . For every K, F ⊆ X, we define the interval [K; F ] = fA 2 P(X): K ⊆ A ⊆ F g. The collection of all intervals [K; F ] where K is finite and F is cofinite is a basis for the product topology on 2X . These intervals are in fact clopen in 2X . In the second section, we study the properties of different types of cube. We study the Cantor space 2N, the Cantor cube 2X and A the cube I . We prove that these cubes are compact, Hausdorff, and T4-spaces. The Cantor cube 2X are second countable, separable and metrizable, and the cube IA where A is countable are second countable, separable and metrizable. In last section, we give more details of the Cantor cube 2X . The starting point of this work is the fact that Alexandroff topologies on X exactly to closed set in 2X .