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رياضة عام تخلفات من ثالثة (نظام جديد) – كلية4 :الفرقة الفصل الدراسي الثاني- شعبة الرياضيات- )التربية(عام م2010-م2011 2011 -6 -11 :تاريخ االمتحان نموذج اجابة ورقة كاملة توبولوجي:المادة – عمرو سليمان محمود – جامعة بنها/ الدكتور:أسم استاذ المادة كلية العلوم – قسم الرياضيات :األسئلة Solve the following (three) questions. First Question: Decide whether each of the following statements is true or false. Justify your answer (by providing a proof, a counter example, a correction, an interpretation). Let ( X , ) be a topological space, the collection of closed sets and A, B X : a) A B A B . b) A is called open domain (or o -set ) iff A Ao . c) A is called G -set iff A Gi ; Gi . i 1 d) Every closed interval [a, b] is an closed set in the natural topology (i.e. [a, b] N ). e) The constant mapping f : ( X , ) (Y , ) : x f ( x) p is a continuous map. Second Question: 1- Let X 1,2,3, : X ,,{1},{2},{1,2} be a topology on X and A {1,3}, B {1,2} . Find the following: a) , Ao , A , B , b( A) and A d b) a base for c) d) e) f) the neighborhood systems N x for all x 1,2,3 a local base (x) for all x 1,2,3 the weight W ( X , ) and the density d ( X , ) . the relative topology A on the set A . 2- Third Question: 1- Let X a, b, c, d , : X , ,{a},{a, b},{a, b, c} be a topology on X and Y x, y, z, w, : Y , ,{x},{ y},{x, y},{y, z, w} be a topology on Y . Consider a map f : X Y defined by f (a ) y f(b) f(d) z, f(c) w and a map g : X Y defined by g(a) g(b) x, g(c) z, g(d) w . Prove that: a) b) c) d) the function the function the function the function f f g g is continuous, is not open, is not continuous. is not open. 2- Define a topological space. Prove that if X is an infinite set, then the collection : G X | G is finite is a topology on the set X . The answer: First Question: a) A B A B (True).: we have A B A A B A A B A B - - - - - - - - - - - - - - - - - - - (1) A B B A B B A A A B A B AB A B B B A B A B (since A B is closed) From (1) and (2) the prove is complete b) A is called open domain (or o -set ) iff A Ao (False): Correction A is called closed domain (or -set ) iff A Ao - - - - - - - -(2) c) A is called G -set iff A Gi ; Gi (False): i 1 Correction A is called G -set iff A Gi ; Gi i 1 d) Every closed interval [a, b] is an closed set in the natural topology (i.e. [a, b] N ) (True) Since [a, b] (a, b) N e) The constant mapping f : ( X , ) (Y , ) : x f ( x) p is a continuous map (True): Since p G f 1 (G) X G 1 p G f (G) Second Question: Let X 1,2,3, : X ,,{1},{2},{1,2} be a topology on X and A {1,3}, B {1,2} . We have: a) : X ,,{2,3},{1,3},{3} Ao {1} A {1,3} A, B X B is dense in X b( A) : A \ Ao {1,3} \ {1} {3} A d : x X | x A \ {x} , so we get: A \ {2} {1,3} A \ {1} {1,3} {1,3} 2 A \ {2} 2 A d A d {3} A \ {3} {1} A \ {3} {1} {1,3} 3 A \ {3} 3 A d A \ {1} {3} A \ {1} {3} {3} 1 A \ {1} 1 A d b) a base for is : X ,{1},{2} c) the neighborhood systems N x for all x 1,2,3 : N1 : X ,{1},{1,2} N 2 : X ,{2},{1,2} N 3 : X , d) a local base (x) for all x 1,2,3 : (1) : {1} (2) : {2} (3) : X e) the weight W ( X , ) : 3 and the density d ( X , ) : B 2 f) the relative topology A {A,,{1}} Third Question: 1) - Let X a, b, c, d , : X , ,{a},{a, b},{a, b, c} be a topology on X and Y x, y, z, w, : Y , ,{x},{ y},{x, y},{y, z, w} be a topology on Y . Consider a map f : X Y defined by f (a ) y f(b) f(d) z, f(c) w and a map g : X Y defined by g(a) g(b) x, g(c) z, g(d) w . We get:: a) the function f is continuous: f 1{Y } f 1{{ x, y, z, w}} X , f 1{} , f 1{x} , f 1{ y} {a} , f 1{x, y} {a} , f 1{ y, z , w} X Hence, f is continuous b) the function f is not open: f is open iff f (G) G f { X } f {{a, b, c, d } Y , f {} , f {a} { y} , f {a, b} { y, z} c) the function g is not continuous. g 1{Y } g 1{{ x, y, z , w}} X , g 1{} , g 1{x} {a, b} , g 1{ y} , g 1{x, y} {a, b} , g 1{ y, z , w} {c, d } Hence, g is not continuous d) the function g is not open: g{ X } g{{a, b, c, d } Y , g{} , g{a} {x} , g{a, b} {x} , g{a, b, c} {x, z} 2) - A topological space is a pair ( X , ) where X is a non-empty set, P( X ) such that 1 X , 2 G1 , G2 G1 G 2 3 {Gi ; i 1,2,...} Gi i1 To prove that : G X | G is finite is a topology on the set X : by definition , X ( finite) X We have G1 , G2 G1, G2 are finite (G1 G2 ) G1 G is finite G1 G2 In fact: {Gi ; i 1,2,...} Gi is finite i Gi Gi is finite i 1 i1 Gi i1 انتهت اإلجابة – عمرو سليمان محمود/الدكتور جامعة بنها – كلية العلوم – قسم الرياضيات