From Metric Space to Topology Prof. Sebastian Vattamattam December 17, 2010Part I METRIC SPACESDenition Let X be a non-empty set. A metric on X is a function d : X X ! R such that, for x; y; z 2 X; (X; d) is called a metric space.Denition Let X be a non-empty set. A metric on X is a function d : X X ! R such that, for x; y; z 2 X; 1. d(x; y) 0 (X; d) is called a metric space.Denition Let X be a non-empty set. A metric on X is a function d : X X ! R such that, for x; y; z 2 X; 1. d(x; y) 0 2. d(x; y) = 0 , x = y (X; d) is called a metric space.Denition Let X be a non-empty set. A metric on X is a function d : X X ! R such that, for x; y; z 2 X; 1. d(x; y) 0 2. d(x; y) = 0 , x = y 3. d(x; y) = d(y; x)(symmetry) (X; d) is called a metric space.Denition Let X be a non-empty set. A metric on X is a function d : X X ! R such that, for x; y; z 2 X; 1. d(x; y) 0 2. d(x; y) = 0 , x = y 3. d(x; y) = d(y; x)(symmetry) 4. d(x; y) d(x; z) + d(z; y)(triangle inequality) (X; d) is called a metric space.Example For x 2 R dene jxj = x if x 0; x if x < 0: Then d : R R ! R dened by d(x; y) = jx yj is a metric(called usual metric) on R and (R; d) is a metric space. It is denoted by (R; j j) also.Example For x = (x1; x2); y = (y1; y2) 2 R2 dene d(x; y) = p(x1 y1)2 + (x2 y2)2 Then d is a metric(called Euclidean Metric) on R2 and (R2; d) is a metric space.Example X 6= ; d : X X ! R dened byd(x; y) = 0 if x = y; 1 if x 6= y: Then, (X; d) is a metric space. d is called the discrete metric.Denition Let (X; d) be a metric space. If a 2 X; r > 0 then dene B(a; r ) := fx 2 X : d(x; a) < rg B(a; r ) is called an open ball with center a and radius r in X: It is also called an open sphere.A set U X is called a neighborhood of a 2 X if 9r > 0 such that B(a; r ) UExample If (X; d) is a discrete metric space, then given a 2 X; 0 < r 1, the ball B(a; r ) = fag Example If (X; d) is a discrete metric space, then given a 2 X; 0 < r 1, the ball B(a; r ) = fag I Let U be any subset of X containing a Example If (X; d) is a discrete metric space, then given a 2 X; 0 < r 1, the ball B(a; r ) = fag I Let U be any subset of X containing a I Then the ball B(a; 12) = fag U Example If (X; d) is a discrete metric space, then given a 2 X; 0 < r 1, the ball B(a; r ) = fag I Let U be any subset of X containing a I Then the ball B(a; 12) = fag U I Therefore any set containing a 2 X is a neighborhood of a:Example In the metric space (R; d), given a 2 R; r > 0, the ball Example In the metric space (R; d), given a 2 R; r > 0, the ball I B(a; r ) = fx 2 R : jx aj < rg Example In the metric space (R; d), given a 2 R; r > 0, the ball I B(a; r ) = fx 2 R : jx aj < rg I = (a r ; a + r ) Example In the metric space (R; d), given a 2 R; r > 0, the ball I B(a; r ) = fx 2 R : jx aj < rg I = (a r ; a + r ) I [a r ; a + r ]; [a r ; a + r ); (a r ; a + r ] Example In the metric space (R; d), given a 2 R; r > 0, the ball I B(a; r ) = fx 2 R : jx aj < rg I = (a r ; a + r ) I [a r ; a + r ]; [a r ; a + r ); (a r ; a + r ] I are neighborhoods of a:Theorem 1:3:2 in [1] If (X; d) is a metric space, a 2 X and U is a neighborhood of a then Theorem 1:3:2 in [1] If (X; d) is a metric space, a 2 X and U is a neighborhood of a then 1. a 2 U Theorem 1:3:2 in [1] If (X; d) is a metric space, a 2 X and U is a neighborhood of a then 1. a 2 U 2. Any superset of U is also a neighborhood of a: Theorem 1:3:2 in [1] If (X; d) is a metric space, a 2 X and U is a neighborhood of a then 1. a 2 U 2. Any superset of U is also a neighborhood of a: 3. The union of a nite collection of neighborhoods of a is also a neighborhood of a:Proof Proof 1. Obvious. Proof 1. Obvious. 2. Let M U: Since U is a neighborhood of a, 9r > o such that B(a; r ) U ) B(a; r ) M Proof 1. Obvious. 2. Let M U: Since U is a neighborhood of a, 9r > o such that B(a; r ) U ) B(a; r ) M 3. Let U1; U2 be two neighborhoods of a 9r1; r2 > 0 such that B(a; r1) U1; B(a; r2) U2 Let r := minfr1; r2g B(a; r ) U1; B(a; r ) U2 ) B(a; r ) U1SU2 Therefore U1SU2 is a neighborhood Open Set If (X; d) is a metric space, a set M X is open if M is a neighborhood of every element in it. That is M is open if for x 2 M; 9r > 0 such that B(a; r ) MExample In the discrete metric space (X; d) every set is open. WHY ?Example In the metric space (R; d) every open interval (a; b) is open. Example In the metric space (R; d) every open interval (a; b) is open. I Let x 2 (a; b) ) a < x < b Example In the metric space (R; d) every open interval (a; b) is open. I Let x 2 (a; b) ) a < x < b I Then x a > 0; b x > 0 Example In the metric space (R; d) every open interval (a; b) is open. I Let x 2 (a; b) ) a < x < b I Then x a > 0; b x > 0 I Let r := min fx a; b xg Example In the metric space (R; d) every open interval (a; b) is open. I Let x 2 (a; b) ) a < x < b I Then x a > 0; b x > 0 I Let r := min fx a; b xg I Then B(x; r ) (a; b) Example In the metric space (R; d) every open interval (a; b) is open. I Let x 2 (a; b) ) a < x < b I Then x a > 0; b x > 0 I Let r := min fx a; b xg I Then B(x; r ) (a; b) I Therefore (a; b) is open.Or Let c = a+b 2 and r = b c Then B(c; r ) = (c r ; c + r ) = (a; b) Therefore (a; b) is open.More Examples from (R; d) The following subsets of R are open. Please try to prove.More Examples from (R; d) The following subsets of R are open. 1. fx 2 Rjx > 0g Please try to prove.More Examples from (R; d) The following subsets of R are open. 1. fx 2 Rjx > 0g 2. fx 2 Rjx < 0g Please try to prove.More Examples from (R; d) The following subsets of R are open. 1. fx 2 Rjx > 0g 2. fx 2 Rjx < 0g 3. R f0g Please try to prove.Why are the following sets not open ? Why are the following sets not open ? 1. fx 2 Rjx 0g Why are the following sets not open ? 1. fx 2 Rjx 0g 2. [1; 2] Why are the following sets not open ? 1. fx 2 Rjx 0g 2. [1; 2] 3. Q Why are the following sets not open ? 1. fx 2 Rjx 0g 2. [1; 2] 3. Q 4. ZExample In the metric space (R2; d) the set U = f(x1; x2)jx1 > 0; x2 > 0g is open. Example In the metric space (R2; d) the set U = f(x1; x2)jx1 > 0; x2 > 0g is open. I Let x = (x1; x2) 2 U Example In the metric space (R2; d) the set U = f(x1; x2)jx1 > 0; x2 > 0g is open. I Let x = (x1; x2) 2 U I Then x1 > 0; x2 > 0 Example In the metric space (R2; d) the set U = f(x1; x2)jx1 > 0; x2 > 0g is open. I Let x = (x1; x2) 2 U I Then x1 > 0; x2 > 0 I Let r = min fx1; x2g Example In the metric space (R2; d) the set U = f(x1; x2)jx1 > 0; x2 > 0g is open. I Let x = (x1; x2) 2 U I Then x1 > 0; x2 > 0 I Let r = min fx1; x2g I Then B(x; r ) U Example In the metric space (R2; d) the set U = f(x1; x2)jx1 > 0; x2 > 0g is open. I Let x = (x1; x2) 2 U I Then x1 > 0; x2 > 0 I Let r = min fx1; x2g I Then B(x; r ) U I Therefore U is open.Example In the metric space (R2; d) the set U = fxj0 < d(x; 0) < 1g is open. Example In the metric space (R2; d) the set U = fxj0 < d(x; 0) < 1g is open. I Let x = (x1; x2) 2 U Example In the metric space (R2; d) the set U = fxj0 < d(x; 0) < 1g is open. I Let x = (x1; x2) 2 U I Then d = d(x; 0) > 0; 1 d > 0 Example In the metric space (R2; d) the set U = fxj0 < d(x; 0) < 1g is open. I Let x = (x1; x2) 2 U I Then d = d(x; 0) > 0; 1 d > 0 I Let r = min fd; 1 dg Example In the metric space (R2; d) the set U = fxj0 < d(x; 0) < 1g is open. I Let x = (x1; x2) 2 U I Then d = d(x; 0) > 0; 1 d > 0 I Let r = min fd; 1 dg I Then B(x; r ) U Example In the metric space (R2; d) the set U = fxj0 < d(x; 0) < 1g is open. I Let x = (x1; x2) 2 U I Then d = d(x; 0) > 0; 1 d > 0 I Let r = min fd; 1 dg I Then B(x; r ) U I Therefore U is open.Example In the metric space (R2; d) the set V = f(x1; x2)ja < x1 < b; c < x2 < dg is open. Example In the metric space (R2; d) the set V = f(x1; x2)ja < x1 < b; c < x2 < dg is open. I Let x = (x1; x2) 2 V Example In the metric space (R2; d) the set V = f(x1; x2)ja < x1 < b; c < x2 < dg is open. I Let x = (x1; x2) 2 V I Then x1 a; b x1; x2 c; d x2 > 0 Example In the metric space (R2; d) the set V = f(x1; x2)ja < x1 < b; c < x2 < dg is open. I Let x = (x1; x2) 2 V I Then x1 a; b x1; x2 c; d x2 > 0 I Let r = min fx1 a; b x1; x2 c; d x2g Example In the metric space (R2; d) the set V = f(x1; x2)ja < x1 < b; c < x2 < dg is open. I Let x = (x1; x2) 2 V I Then x1 a; b x1; x2 c; d x2 > 0 I Let r = min fx1 a; b x1; x2 c; d x2g I Then B(x; r ) V Example In the metric space (R2; d) the set V = f(x1; x2)ja < x1 < b; c < x2 < dg is open. I Let x = (x1; x2) 2 V I Then x1 a; b x1; x2 c; d x2 > 0 I Let r = min fx1 a; b x1; x2 c; d x2g I Then B(x; r ) V I Therefore V is open.Theorem A; p:60 in [2] In a metric space (X; d); the empty set and the whole space X are open.Proof Proof 1. Obvious Proof 1. Obvious 2. For a 2 X and any r > 0; B(a; r ) X:Theorem B, p:61 in [2] In metric space (X; d), each open sphere is open.Proof Proof I Let for r > 0; B(a; r ) = fx 2 Xjd(x; a) < rg. Proof I Let for r > 0; B(a; r ) = fx 2 Xjd(x; a) < rg. I Take b 2 B(a; r ) ) d(b; a) < r Proof I Let for r > 0; B(a; r ) = fx 2 Xjd(x; a) < rg. I Take b 2 B(a; r ) ) d(b; a) < r I Let d = d(b; a) and r1 = min fd; r dg Proof I Let for r > 0; B(a; r ) = fx 2 Xjd(x; a) < rg. I Take b 2 B(a; r ) ) d(b; a) < r I Let d = d(b; a) and r1 = min fd; r dg I Then B(b; r1) B(a; r ) Proof I Let for r > 0; B(a; r ) = fx 2 Xjd(x; a) < rg. I Take b 2 B(a; r ) ) d(b; a) < r I Let d = d(b; a) and r1 = min fd; r dg I Then B(b; r1) B(a; r ) I Therefore B(a; r ) is open.Theorem C, p:61 in [2] Let (X; d) be a metric space. A set M X is open , it is a union of open balls.Proof Proof 1. Suppose M is open. I Let a 2 M Proof 1. Suppose M is open. I Let a 2 M I 9r > 0 such that B(a; r ) M 2. Suppose M = SfB(a; r )ja 2 Mg I x 2 M ) x 2 B(a; r ) for some r > 0 and a 2 M I Let r1 := r d(x; a) > 0 Proof 1. Suppose M is open. I Let a 2 M I 9r > 0 such that B(a; r ) M I Then M = SfB(a; r )ja 2 Mg, a union of open balls. 2. Suppose M = SfB(a; r )ja 2 Mg I x 2 M ) x 2 B(a; r ) for some r > 0 and a 2 M I Let r1 := r d(x; a) > 0 I Then B(x; r1) B(a; r ) M Proof 1. Suppose M is open. I Let a 2 M I 9r > 0 such that B(a; r ) M I Then M = SfB(a; r )ja 2 Mg, a union of open balls. 2. Suppose M = SfB(a; r )ja 2 Mg I x 2 M ) x 2 B(a; r ) for some r > 0 and a 2 M I Let r1 := r d(x; a) > 0 I Then B(x; r1) B(a; r ) M I Therefore M is open.Theorem D, p:61 in [2] Let (X; d) be a metric space. Then, Theorem D, p:61 in [2] Let (X; d) be a metric space. Then, 1. Any union of open sets in X is open. Theorem D, p:61 in [2] Let (X; d) be a metric space. Then, 1. Any union of open sets in X is open. 2. Any nite intersection of open sets in X is openProof(1) Let fUj2 Jg be an arbitrary family of open sets in X; and U := SfUj2 Jg To prove that U is open. Proof(1) Let fUj2 Jg be an arbitrary family of open sets in X; and U := SfUj2 Jg To prove that U is open. 1. x 2 U ) x 2 U; for some 2 J Proof(1) Let fUj2 Jg be an arbitrary family of open sets in X; and U := SfUj2 Jg To prove that U is open. 1. x 2 U ) x 2 U; for some 2 J 2. Since Uis open, Uis a neighborhood of x. Proof(1) Let fUj2 Jg be an arbitrary family of open sets in X; and U := SfUj2 Jg To prove that U is open. 1. x 2 U ) x 2 U; for some 2 J 2. Since Uis open, Uis a neighborhood of x. 3. But U Uand hence U is a neighborhood of x. Proof(1) Let fUj2 Jg be an arbitrary family of open sets in X; and U := SfUj2 Jg To prove that U is open. 1. x 2 U ) x 2 U; for some 2 J 2. Since Uis open, Uis a neighborhood of x. 3. But U Uand hence U is a neighborhood of x. 4. Therefore, U is open.Proof(2) Let fUi ji 2 f1; 2; ::; ngg be a nite family of open sets in X; and U := TfUi ji 2 f1; :::; ngg To prove that U is open. Proof(2) Let fUi ji 2 f1; 2; ::; ngg be a nite family of open sets in X; and U := TfUi ji 2 f1; :::; ngg To prove that U is open. 1. x 2 U ) x 2 Ui ; 8i Then for each i ; Ui is a neighborhood of x. Proof(2) Let fUi ji 2 f1; 2; ::; ngg be a nite family of open sets in X; and U := TfUi ji 2 f1; :::; ngg To prove that U is open. 1. x 2 U ) x 2 Ui ; 8i Then for each i ; Ui is a neighborhood of x. 2. Therefore for each i ; 9ri > 0 such that B(x; ri ) Ui Proof(2) Let fUi ji 2 f1; 2; ::; ngg be a nite family of open sets in X; and U := TfUi ji 2 f1; :::; ngg To prove that U is open. 1. x 2 U ) x 2 Ui ; 8i Then for each i ; Ui is a neighborhood of x. 2. Therefore for each i ; 9ri > 0 such that B(x; ri ) Ui 3. Let r := minfri ; i 2 f1; :::; ngg Proof(2) Let fUi ji 2 f1; 2; ::; ngg be a nite family of open sets in X; and U := TfUi ji 2 f1; :::; ngg To prove that U is open. 1. x 2 U ) x 2 Ui ; 8i Then for each i ; Ui is a neighborhood of x. 2. Therefore for each i ; 9ri > 0 such that B(x; ri ) Ui 3. Let r := minfri ; i 2 f1; :::; ngg 4. B(x; r ) B(x; ri ); 8i ) B(x; r ) U Proof(2) Let fUi ji 2 f1; 2; ::; ngg be a nite family of open sets in X; and U := TfUi ji 2 f1; :::; ngg To prove that U is open. 1. x 2 U ) x 2 Ui ; 8i Then for each i ; Ui is a neighborhood of x. 2. Therefore for each i ; 9ri > 0 such that B(x; ri ) Ui 3. Let r := minfri ; i 2 f1; :::; ngg 4. B(x; r ) B(x; ri ); 8i ) B(x; r ) U 5. Therefore U is open.Part II TOPOLOGICAL SPACESDenition A collection of subsets of a set X is a topology if The elements of are called open sets in X and (X; ) is called a topological space.Denition A collection of subsets of a set X is a topology if 1. X; 2 ; The elements of are called open sets in X and (X; ) is called a topological space.Denition A collection of subsets of a set X is a topology if 1. X; 2 ; 2. any union of sets in is in , and The elements of are called open sets in X and (X; ) is called a topological space.Denition A collection of subsets of a set X is a topology if 1. X; 2 ; 2. any union of sets in is in , and 3. any nite intersection of sets in is in The elements of are called open sets in X and (X; ) is called a topological space.Example Let X be a set and = P(X); the power set of X. Then (X; ) is a topological space called the discrete space.Example Let X be a set and = f; Xg: Then the topological space (X; ) is called the indiscrete space.Theorem Every metric space (X; d) is a topological space, with = fU XjU is open in the metric senseg is called the metric topology induced by the metric d: If (X; d) is a discrete metric space, then the metric topology is the discrete topology.Example Let be the family of open sets in the metric space (R; j j): Then (R; ) is a topological space. is called the usual topology on R:Denition A topological space (X; ) is metrizable if the topology is induced by a metric.THANK YOU Sebastian Vattamattam vattamattam@gmail.comW. Beekmann,Analysis 2,Fern University in Hagen George F. Simmons, Topology and Modern Analysis,International Student Edition, McGraw-Hill, 1963.