Basic Topology - CHAPTER 1 BASIC TOPOLOGY Topology,...

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Unformatted text preview: CHAPTER 1 BASIC TOPOLOGY Topology, sometimes referred to as the mathematics of continuity, or rubber sheet geometry, or the theory of abstract topological spaces, is all of these, but, above all, it is a language , used by mathematicians in practically all branches of our science. In this chapter, we will learn the basic words and expressions of this language as well as its grammar, i.e. the most general notions, methods and basic results of topology. We will also start building the library of examples, both nice and natural such as manifolds or the Cantor set, other more complicated and even pathological. Those examples often possess other structures in addition to topology and this provides the key link between topology and other branches of geometry. They will serve as illustrations and the testing ground for the notions and methods developed in later chapters. 1.1. Topological spaces The notion of topological space is deFned by means of rather simple and abstract axioms. It is very useful as an umbrella concept which al- lows to use the geometric language and the geometric way of thinking in a broad variety of vastly different situations. Because of the simplicity and elasticity of this notion, very little can be said about topological spaces in full generality. And so, as we go along, we will impose additional restric- tions on topological spaces, which will enable us to obtain meaningful but still quite general assertions, useful in many different situations in the most varied parts of mathematics. 1.1.1. Basic defnitions and frst examples. D EINITION 1.1.1. A topological space is a pair ( X, T ) where X is a set and T is a family of subsets of X (called the topology of X ) whose elements are called open sets such that (1) , X T (the empty set and X itself are open), (2) if { O } A T then A O T for any set A (the union of any number of open sets is open), (3) if { O i } k i =1 T , then k i =1 O i T (the intersection of a Fnite number of open sets is open). 5 6 1. BASIC TOPOLOGY If x X , then an open set containing x is said to be an ( open ) neigh- borhood of x . We will usually omit T in the notation and will simply speak about a topological space X assuming that the topology has been described. The complements to the open sets O T are called closed sets . E XAMPLE 1.1.2. Euclidean space R n acquires the structure of a topo- logical space if its open sets are deFned as in the calculus or elementary real analysis course (i.e a set A R n is open if for every point x A a certain ball centered in x is contained in A ). E XAMPLE 1.1.3. If all subsets of the integers Z are declared open, then Z is a topological space in the socalled discrete topology....
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This note was uploaded on 02/07/2011 for the course PHYS 101 taught by Professor Aster during the Spring '11 term at East Tennessee State University.

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Basic Topology - CHAPTER 1 BASIC TOPOLOGY Topology,...

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