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Unformatted text preview: Chapter 2 Topological Spaces This chapter contains a very bare summary of some basic facts from topology. 2.1 Definition of Topology A topology O on a set X is a collection of subsets of X satisfying the following conditions: (T1) ∅ ∈ O (T2) X ∈ O (T3) O is closed under finite intersections, i.e. if A, B ∈ O then A ∩ B ∈ O (T4) O is closed under unions, i.e. if S ⊂ O then ∪ S ∈ O We have used the settheoretic notation for unions: ∪ S is the union of all the sets in S . Note also that by the usual induction argument, condition (T2) implies that if A 1 , ..., A n are a finite number of open sets then A 1 ∩ ··· ∩ A n is also open. A topological space ( X, O X ) is a set X along with a topology O X on it. Usuallly, we just say “ X is a topological space.” If x ∈ X and U is an open set with x ∈ U then we say that U is a neighborhood of x . There are two extreme topologies on any set X : • the indiscrete topology {∅ , X } 1 2 CHAPTER 2. TOPOLOGICAL SPACES • the discrete topology P ( X ) consisting of all subsets of X A set A in O is said to be open in the topology O . The complement of an open set is called a closed set. Taking complements of (T1)(T4) it follows that: (C1) X is closed (C2) ∅ is closed (C3) the union of a finite number of closed sets is closed (C4) the intersection of any family of closed sets is closed. Let F ⊂ X . The interior F of F is the union of all open sets contained in F ( F could be empty, in case the only open set which is a subset of F is the empty set). It is of course an open set and is the largest open set which is a subset of F . The closure F of F is the intersection of all closed sets which contain F as a subset. Thus F is a closed set, the smallest closed set which contains F as a subset. The intersection of any set of topologies on X is clearly also a topology. Let S be any nonempty collection of subsets of X . Consider the collection T S of all topologies which contain S , i.e. for which the sets of S are open. Note that P ( X ) ∈ T S . Then ∩ T S is the smallest topology on X containing all the sets of S . It is called the topology on X generated by S . A topological space X is Hausdorff if any two points have disjoint neigh borhoods: i.e. for any x, y ∈ X with x 6 = y , there exist open sets U and V , with x ∈ U , y ∈ V , and U ∩ V = ∅ . 2.2 Continuous maps Let ( X, O X ) and ( Y, O Y ) be topological spaces, and f : X → Y a mapping. We say that f is continuous if f 1 ( O Y ) ⊂ O X , 2.3. COMPACT SETS 3 i.e. if for every open set B ⊂ Y the inverse image f 1 ( A ) is an open subset of X . It is clear that the identity map X → X is continuous, when X is given any particular topology....
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This note was uploaded on 01/27/2010 for the course MATH 7312 taught by Professor Sengupta during the Spring '02 term at LSU.
 Spring '02
 Sengupta
 Logic, Topology, Sets

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