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Topology (Part 1) Original Notes adopted from April 2, 2002 (Week 25) © P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics typed by A. Ku Ong R, A subset S of R is open if S is a union of open intervals; ie) whenever x ∈ S , there exists open interval (a,b) such that x ∈ (a,b) & (a,b) ⊂ S. 7 -2.1 3 7 9 13 17 S = { x: x< -7 or x ∈ (-2.1, 3) or x ∈ (7,9) or x ∈ (13,17)} In R2, an open disk is a set of form {(x,y): (x-a) 2 + (y-b) 2 < r} ie) an integer of circle. R2, A subset S of R2 is open if S is a union of open disks; ie) whenever (x,y) ∈ S , there exists open disk D such that x ∈ D & D ⊂ S. eg. {(x,y): y > x2 } Note: (in R and R2 ): the union of any number of open sets is open. Eg. Let In = (-1/n, 1/n) ∞ ∩In = {0} not open. n=1 The intersection of open sets need not be open. The intersection of 2 open sets is open. Let S1 ,S2 be open and suppose x ∈ S1 ∩ S2 there exists, (a1 ,b1 ) such that x ∈ (a1 ,b1 ) & (a1 ,b1 ) ⊂ S1 there exists (a2 ,b2 ) such that x ∈ (a2 ,b2 ) & (a2 ,b2 ) ⊂ S2 a1 a2 x b1 (a2 ,b1 ) ⊂ S1 ∩ S2 (S1 ∩ S2) ∩ S3 We define ∅ to be open. b2 For R & R2 , we have: 1) ∅ & whole space are open. 2) The union of any number of open sets is open 3) The intersection of any finite number of open sets is open. A topological space is a set together with a collection of surds, called the open sets, such that the union of any number of open sets is open, the intersection of a finite number of open sets is open, and the empty set and the entire space are both open. EXAMPLES OF TOPOLOGICAL SPACES. 1) R , R2 with their usual topologies, as above. 2) Let X = {a,b,c} and let every subset of X be designated as "open". 3) Take set R and designate a subset S as "open" if R\S (complement) is countable or if S = ∅. S1 , S2 .... R\ S1 ,R\ S2 countable R\ S1 U R\ S2 Let x be any set with subsets designated "open" satisfying 3 properties, x is a topological space. Definition: A subset F is closed if X\F is open. Definition: A point x is a limit point of a subset S if whenever U is an open set containing x, U contains a point of S other than x (x may or may not be in S). Theorem: A set is closed if and only if it contains all its limit points. Proof: Suppose S is closed. Then X\S is open. If x ∈ X\S , then x is not a limit point since X\S is an open set whose intersection with S is empty. ∴ all limit points are in S, not X\S. Conversely, (if it contains all its limit points its closed). Suppose S contains all its limit points. Show: X\S open. Let x ∈ X\S x not a limit point of S, so there exists Ux , Ux ∩ S = ∅. Ux ⊂ X \S. X\S = U Ux x∈ X\S X\S is a union of open sets, so X\S is open Definition: If f: X→ Y, and S ⊂ Y, then f--1(S) = {x ∈ X: f(x) ∈ S} ("The inverse image of S"). Definition: If X and Y are topological spaces, & f: X →Y, then f is continuous if f- -1(U) is an open subset of X whenever U is an open subset of Y. Note: For f: R →R, above agrees with usual definition. ∀ ∈>0, there exists δ >0 such that |f (x) – f(a) | < ∈ whenever |x-a| < δ _ Definition: The closure of a subset S of a topological space is S U {limit points of S} denoted by S. Eg. In R with usual topology. _ 1) If S = (3,9), S = [3,9] 2)S = {1/n: n =1,2,3 ...} 0 is a limit point. 1 = S U {0} Definition: Homeomorphism is a one-to-one function from a topological space onto another topological space which is continuous and which has a continuous inverse. Two topological spaces are said to be homeomorphic if there is a homemorphism from one onto the other. Examples: [-Homeomorphic] [Not Homeomorphic] Disconnected ______ _______ Definition: The topological space X is disconnected if there exists subsets A & B of X both of which are closed, neither of which is empty, such that X = AUB & A ∩ B =∅. Then A & B are both open too. In every topological space, X& Y are both open and closed. A space is disconnected if and only if it has a subset 0 in addition to ∅ & x that is both open & closed.