A Study of New Topologies on R Thesis Submitted in partial fulfillment of the requirements for the award of degree of Masters of Science in Mathematics and Computing Submitted by Sakshi Reg. No.- 301403015 Under the guidance of Dr. Jatinderdeep kaur School of Mathematics Thapar University Patiala-147004(PUNJAB) INDIA July, 2016 ABSTRACT The present dissertation entitled, \A Study of New Topologies on R" comprises certain investigations carried out by me at School of Mathematics, Thapar Uni- versity, Patiala, under the supervision of Dr. Jatinderdeep Kaur, Assistant Professor, School of Mathematics, Thapar University, Patiala. The aim of this work is to find new topologies with this new set of basis B− and B+. The topologies generated from this new set of basis are different from stan- dard, lower limit, upper limit and k-topologies. We compared well known topologies with new topologies on R. The whole work is divided into four chapters. Chapter 1 is introduction which includes brief account of definitions and their examples which will be required for the later chapters. In chapter 2, we have studied some main theorems which guar- antees how topology can be generated from given base and vice-versa. The aim of chapter 3 is to study well known topologies on R and their topological properties. In chapter 4, we introduced new topologies on real line and compared them with well known topologies on R. At the end of this dissertation, we have bibliography of research papers and books cited in the dissertation. Contents 1 Introduction to Topology 2 1.1 Introduction . 2 1.2 Basic concepts and definitions . 3 2 Bases for Topology 9 2.1 Introduction . 9 2.2 Main results . 10 2.3 Examples to illustrate main results . 12 3 Some known Topologies on Real Line R and their Properties 14 3.1 Properties of standard topology . 15 3.2 Properties of lower limit topology . 18 4 New Topologies on R and Their Properties 22 4.1 Introduction . 22 4.2 New topologies on R ........................... 22 4.3 Comparison of well known topologies on R ............... 24 4.4 Comparison of new topologies with well known topologies on R ... 26 4.5 Conclusion . 27 1 Chapter 1 Introduction to Topology 1.1 Introduction 'Topology' is the area of mathematics which studies continuity and related concepts. Important fundamental notions like open sets, closed sets, limit points and continu- ity are introduced. Originally coming from questions in analysis and differential geometry. L.Euler in 1736, the paper entitled, \seven bridges of konigsberg"[2] is regarded as one of important study in modern topology. In this paper, he pointed out that the choice of route inside each land mass is irrelevant. The only useful property of a route is the sequence of bridges crossed, he recognized that the key information was the number of bridges and the list of their end points which typically development of topology. It is difficult to fix a date for the starting of topology. The first time use of term Topology that we know of appeared in title of a book written by J.W. Listing in 1847. Further, many authors like Cauchy, Riemann and Enrico Betti studied various results in topology. In present time topology is an important branch of pure mathematics. Topo- logical ideas are present in almost all areas of today's mathematics. The subject of topology itself consists of several different branches such as point set topology, algebraic topology and differential topology, which have relatively little in common 2 Topology can be defined as \the study of qualitative properties of certain ob- jects that are invariant under a certain kind of transformation". The motivation insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way the are put together. The square and the circle have many properties in common: they are both same dimen- sional objects and both separate the plane into two parts: inside and outside part. Basic topology is concerned with boundaries . What is an open set? When is a set \in one piece"? When can we \put a set" inside a box"? What is a \boundary"? What is a "neighborhood"? And so on. 1.2 Basic concepts and definitions In this section, some basic definitions are presented. Here, we give a brief account of definitions and results which will be required for the later chapters. However, some of the definitions and notations will be repeated occasionally in various chap- ters for the sake of convenience. Definition 1.1 ([5],p.76) A topology on a set X is a collection T of subsets of X having the following properties: (i) ϕ and X are in T . (ii) The union of the elements of any subcollection of T is in T . (iii) The intersection of the elements of any finite subcollection of T is in T If T is topology on X, then the pair (X; T ) is called Topological space. The elements of T are called open sets. A subset F ⊆ X is called closed sets if its compliment is open. Remark 1.2 Any finite intersection U1 \ U2::: \ Un of open sets is open. It is impor- tant to note that it is in general not true that an arbitrary(infinite) union of open 3 sets would be open. Remark 1.3 Open and closed sets are not mutually exclusive. Infact, subsets that are both open and closed often exist. Example 1.4 (Discrete Topological space) Let X be arbitrary set, if we consider T as the collection of every subset of X, then (X; T ) is called discrete topological space. Example 1.5 (Indiscrete Topology) Let X be a non-empty set, the collection consisting of X and ϕ only is also a topology on X, it is called the indiscrete topology. These two examples are easy to understand. However, some more examples of important topological spaces are given below: Example 1.6 Let X = R, Consider the collection T = fU ⊆ RjU is the union of open intervalsg where open interval is defined as (a; b) = fx 2 Rja < x < bg with a; b 2 R. This topology is called Standard or Euclidean topology on R. Example 1.7 (Co-Finite Topology) Let X be a non-empty set, the collection of subsets of X consisting of the empty set together with those non-empty subsets of X whose complements are finite is called Co-finite Topology. Example 1.8 (Co-Countable Topology) Let X be a non-empty set, the collection of subsets of X consisting of the empty and all those non-empty subsets of X whose compliments are countable is called Co-countable Topology. 4 Definition 1.9 Suppose T1 and T2 are two topologies on the set X and if T1 ⊆ T2, then T2 is Finer Topology than T1 or T1 is coarser than T2 and if T1 ⊂ T2 then T2 is strictly finer than T1. T1 is comparable with T2 if either T1 ⊆ T2 or T2 ⊆ T1. Remark 1.10 Two topologies on X need not be comparable. For example Let X=fa,b,cg, and two topologies on X are T1= fϕ,fag,Xg and T2= fϕ,fbg,Xg. It can be easily noted that neither T1 * T2 nor T2 * T1, then T1 and T2 are not comparable. Definition 1.11 ([5],p.78) If X is a set, a basis for a topology on X is a collection B of subsets of X such that: (i) For each x 2 X, there is at least one basis element B containing x. (ii) If x belongs to the intersection of two basis elements B1 and B2, then there is T a basis elements B3 containing x such that B3 ⊂ B1 B2. Example 1.12 If X is any set, the collection of all one-point subsets of X is a basis for the discrete topology on X. Example 1.13 The collection of all open intervals in the real line forms a base for the standard topology. Definition 1.14 ([4],p.171) Let X be a topological space. A class B of open subsets of X is said to be an open base for X if every open set in X is a union of members of B. Example 1.15 The class of open intervals form an open base for the topological space R of real numbers with usual topology. Definition 1.16 ([4],p.172) Let (X; T ) be a topological space and x 2 X. Then the class Bx of open sets containing points x 2 X is local base if for each open set G 5 containing x, there exists Gx 2 Bx such that x 2 Gx ⊂ G. Example 1.17 Let (R;U) be a topological space and let x 2 R. Then the collection of Bx = (x − ϵ, x + ϵ) : 0 < ϵ 2 R forms a local base at x. Definition 1.18 ([4],p.178) Let (X; T ) be a topological space. The space X is said to be First countable space if for all x 2 X, there exists a countable class Bx of open sets containing x such that every open set G containing x also contains a member of Bx. Example 1.19 Every discrete topology (X; D) is first countable. Definition 1.20([5],p.190) If a space X has a countable basis for its topology, then X is said to be second countable space. Example 1.21 The topological space R of real numbers with usual topology is second countable.