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THREE DIMES of TOPOLOGY A. Candel Class Notes for Math 262, Winter 95-96, the University of Chicago. Preface Chapter I. Topologi

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THREE DIMES OF TOPOLOGY A. Candel Class Notes for Math 262, Winter 95-96, The University of Chicago. Preface Chapter I. Topological spaces 1. The concept of topological space 2. Bases and subbases 3. Subspaces, unions and hyperspaces 4. Pseudometric and metric spaces 5. The order topology 6. More examples Chapter II. Continuous maps 1. Continuous maps 2. Continuity and convergence Chapter III. Product and quotient spaces 1. The product topology 2. Quotient spaces 3. Cutting and pasting Chapter IV. Connected and path-connected spaces 1. Connected spaces 2. Path-connected spaces 3. Local connectedness and path-connectedness Chapter V. Convergence 1. Hausdor® spaces 2. Countability axioms 3. Filters 4. Ultra¯lters Chapter VI. Compact spaces 1. Compact spaces 2. The Tichonov theorem 3. Local compactness 4. Compacti¯cation 5. Components of a compact hausdor® space Chapter VII. The Urysohn lemma 1. Regular and normal spaces Typeset by AMS-TEX 1 2 TOPOLOGY 2. Urysohn and Tietze lemmas 3. Embeddings 4. Stone-Cec· h compacti¯cation 5. Metrization Chapter VIII. Paracompact spaces 1. Paracompact spaces 2. Partitions of unity Chapter IX. Function spaces 1. Topologies on function spaces 2. The evaluation map 3. Uniform spaces 4. Ascoli's theorem Chapter X. Topological groups 1. Topological groups 2. Topological vector spaces 3. Haar measure 4. Stone-Weierstrass theorem 5. Topological spaces and Banach algebras Homeworks and exams Bibliography NOTE: These notes are being o®ered without warranty. Bibliography and references have not been collected. Chapter X does not exist. 1. THE CONCEPT OF TOPOLOGICAL SPACE 1 CHAPTER I TOPOLOGICAL SPACES 1. The Concept of Topological Space De¯nition. A topological space is a pair (X; O) consisting of a set X and a collection O of subsets of X (called \open sets"), such that the following axioms hold: (1) Any union of open sets is an open set. (2) The intersection of any two open sets is open. (3) Both ¿ and X are open sets. One also says that O is the topology of the topological space (X; O), and usually we will drop O and speak of a topological space X. De¯nition. Let X be a topological space. (1) A subset F of X is called closed if X n F is open. (2) A subset N of X is called a neighborhood of x 2 X if there is an open set U such that x 2 U ½ N. (3) Let Y be a subset of X. A point x in X is called an interior, exterior or boundary point of Y if Y , X n Y or neither is a neighborhood of X. (4) The set Y ± of the interior points of Y is called the interior of Y . (5) The set Y ¡ of points of X which are not exterior points of Y is called the closure of Y . Exercise. The interior (closure) of a set is the largest open set (smallest closed set) contained in it (which contains it). The duality open-closed allows us to de¯ne a topology in terms of closed sets. The axioms are obtained from the ones above by means of Morgan's laws. Axioms for closed sets. A topological space is a pair (X; C) consisting of a set X and a family of subsets of X (called \closed sets") such that (1) Any intersection of closed sets is closed. (2) The union of any two closed sets is closed. (3) The empty set and X are closed sets. Originally the notion of topology was de¯ned in terms of neighborhoods. Axioms for Neighborhood. A topological space is a pair (X; N) consisting of a set X and a family N = fNxgx2X of sets Nx of subsets of X (called \neighborhoods") such that: (1) Each neighborhood of x contains x, and X is a neighborhood of each of its points. (2) If N ½ X contains a neighborhood of x, then N itself is a neighborhood of x. (3) The intersection of two neighborhoods of x is a neighborhood of x. (4) Each neighborhood of x contains a neighborhood of x which is also a neighborhood of each of its points. Typeset by AMS-TEX 2 TOPOLOGY Kuratowski Closure axioms. A topological space is a pair (X;¡ ) consisting of a set X and a map ¡ : PX ! PX from parts of X into itself such that: (1) ¿¡ = ¿. (2) A ½ A¡ for all A ½ X. (3) A¡¡ = A¡. (4) (A [ B)¡ = A¡ [ B¡. Exercise. Formulate what the equivalence of this notions means and prove it. Exercise. Kuratowski closure axioms can be replaced by only two: (1) ¿¡ = ¿ and (2) A [ A¡ [ B¡¡ ½ (A [ B)¡. 2. Bases and subbases In several of the examples that we will discuss we see that we de¯ne open sets as union of a smaller collection of open sets. This is an important concept. Base. Let X be a topological space. A collection B of open sets is called a basis for the topology if every open set can be written as a union of sets in B. A related concept is the following. Subbase. Let X be a topological space. A collection S of open sets is a subbasis for the topology if every open sets is a union of ¯nite intersections of elements of S. The collection of all open sets of a topological space is both a base and a subbase. Of course one is usually interested in smaller families. The importance of this de¯nitions is the following. Proposition. Let X be a set and let B be a collection of subsets of X whose union is X and such that for any B, B0 in B and x 2 B \ B0 there is B00 2 B such that x 2 B00 ½ B \ B0. Then there is exactly one topology O(B) on X for which B is a basis. Proposition. Let X be a set and let S be an arbitrary collection of subsets of X. Then there is exactly one topology O(S) on X for which S is a subbasis. The only property that requires a comment is that ¿ and X are in O(S). This hold by using the natural convention that the intersection of an empty family of sets is the whole space, the union of an empty family of sets is the empty set. One has to read the meaning of x 2 \i2I Bi as x 2 Bi for every i 2 I, and of x 2 [i2I Bi as x 2 Bi for some i 2 I. These theorems are important because we usually want topologies satisfying certain properties, and we want these topologies to be as smaller as possible. The notion of \ O is smaller than O0 " refers to the partial relation O ½ O0 (the usual name is \coarser") or O0 is ¯ner that O. There is a coarsest topology, namely the trivial one, and a ¯nest one, also called discrete. In the typical situation the desired topology should be as coarse as possible, and contain at least the elements of S. Neighborhood base. A neighborhood base at x in the topological space X is a subcollection Bx of the neighborhood system Nx, having the property that each N 2 Nx contains some V 2 Bx. That is Nx is determined by: Nx = fN ½ X; V ½ N for some V 2 Bxg: For example, in any topological space the open neighborhoods of x form a neighborhood base at x. In a metric space, the balls centered at x with rational radius form a neighborhood base at x. 3. SUBSPACES, UNIONS, AND HYPERSPACES 3 De¯nition. A topological space in which every point has a countable neighborhood base is said to be ¯rst countable. It is said to be second countable if it has a countable base for its topology. A topology can also be described by giving a collection of basic neighborhoods at each point. Theorem. Let X be a topological space and for each x 2 X let Bx be a neighborhood base at x. Then: (1) if V 2 Bx, then x 2 V , (2) if U; V 2 Bx, there exists W 2 Bx such that W ½ U \ V , (3) if V 2 Bx, there is some V0 2 Bx such that if y 2 V0, then there is some W 2 By with W ½ V , and (4) U ½ X is open if and only if it contains a basic neighborhood of each of its points. Conversely, given a set X and a collection of subsets Bx of X assigned to each of its points so as to satisfy (1), (2), (3) above, there is a topology on X whose open sets are de¯ned by (4) and which has Bx as neighborhood base of each of its points. Rather than giving a proof of this, we describe some examples. The Moore plane. Let M denote the closed upper half plane (x; y), y ¸ 0. For a point in the open upper half plane basic neighborhoods would be the usual open discs (taken small enough so that they lie in M). For a point z in the x-axis the basic neighborhoods would be the sets fzg [ B, where B is an open ball in the upper half plane tangent to the x-axis at z. The slotted plane. At each point x in the plane, the basic nhoods at x are the sets x [ B, where B is an open ball about x with a ¯nite number of straight lines through x removed. The looped line. At each point x 6= 0 of the real line the basic neighborhoods would be the open intervals centered at x.
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