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Math. 373 Problems Sheet 1 Q1: If A is a subset of X, show that the family of all subsets of X which contains A, together with the empty set , is a topology on X. what topology results when A = X ? when A = ? Q2: Show that the intersection of a family of topologies on X is a topology on X. Q3: Let Z + be the set of positive integers . For each positive integer n let Un = {n, n + 1, n + 2,...}. Let T = { ,U1,U2,U3,...}. (i) Show that T is a topology on Z . (ii) List the open sets containing the positive integer 7. Q4: Prove that T is the discrete topology on X if and only if every singleton in X is an open set. Q5: Show that if X is a finite set, then the discrete topology on X coincide with the cofinite topology on X. Q6:(a)Is there a set upon which the discrete and indiscrete topologies are equal ? (b) Give an example of a topology on an infinite set which has only a finite number of elements. (Don't use indiscrete topology). Q7: (a) Let X be a non empty set. Define a subset U of X such that X-U is countable, let T = { X, , all such U }.Show that T is a topology on X. This topology is called cocountable topology. (b) If X is countable, describe the topology . Q8: Prove that the intersection of a finite number of open sets in topological space is open set. Q9: Give an example of two topologies on a set X such that their union is not a topology on X. Q10: Give an example of a collection of open sets in a space X whose intersection is not open. Q11: Let T be a topology on X consisting of four elements, i.e. T= {X, , A, B} where A and B are non empty distinct proper subset of X. What conditions must A and B satisfy ? Q12: Let La = (a, ), where a R , let T= { R , , all La} prove that T is a topology on R. Q13: Let R be the usual top. And let p be a point not in R. Let R R p. Define a collection T of subsets of R to be of two types: Type1: U , where U is open subset of R. Type 2: R B , where B is closed and bounded subset of R. Show that T is a topology on R . Q14: Let U be a collection of subsets of X . Suppose that and X are in U , and that finite unions and arbitrary intersections of elements of U are in U. Show that the collection T = { X- C : C U } is a topology on X. Q15: Show that if A is closed in B and B is closed in X, then A is closed in X. Q16: Show that if U is open in X and A is closed in X, then U A is open in X, and A U is closed in X. Q17: Give an example of a collection of closed sets whose union is not closed. Q18:Let R have the L.R.T. Describe what kind of sets are closed in this space ? Do the same for the cofinite topology on R. Q19: In R, do the rationales form an open set ? Closed set ? Neither ? Both ? Prove your assertion? Q20: Consider the set of real numbers A = { x : 0 < x < 1} {2,3} in R. Describe A for the following topologies on R: (a) Usual. (b) Cofinite. (c) L.R.T. (d) Discrete. Q21: Give an example to show that A B A B . MATH 373 PROBLEMS SHEET 2 Q1: Consider R and the set A = {x-1 R: 0 < x 2} {5,7}. Describe Int(A), Ext(A), Bd(A), A, A if the topology on R is the (a) Usual (b) Cofinite (c) Discrete (d) Indiscrete (e) L.R.T. Q2: Let A and B be subsets of a space X. Prove the following: (a) Int(A) = A Bd(A) (b) Bd(A) = A Int(A) (c) Int(A) Int(B) Int(A B), give an example to show that the converse is not true. (d) Suppose that Bd(A) Bd(B) = . Prove that the converse of (c) is true in this case. Q3: Let A be a subset of a topological space X, prove that A is closed in X if and only if A contains all of its limit points. Q4: Prove: If A is any subset of a closed set B, then A B . Q5: Show by a counter example that the function f which assigns to each subset A its interior, i.e, f A Int A, does not commute with the function which assigns to each set its closure, i.e, g A A . i.e, show that gf fg . Q6: Let T1 and T2 be two topologies on a set X such that T1 T2 , let A X . (i) Show that every limit point of A with respect to T2 is also a limit point of A with respect to T1 . (ii) Construct an example in which a limit point of A with respect to T1 is not a limit point of A with respect to T2 . Q7: Let A and B be subsets of a topological X . Then A B A B . Q8: Definition: “ Let A be a subset of a topological space X, a point x X is called an isolated point of A iff there is an open set G in X containing x such that G A= {x}”. Prove that if X having no isolated points, then every open subset of X can have no isolated points. Q9: Prove that if x is a limit point of A R and U is an open set in R containing x, then U A is an infinite set. Q10: The set A is called perfect iff A A . Prove that a set A is perfect iff it is closed and has no isolated points. Q11: Prove : (i) Bd(A) A = iff A is open. (ii) Bd(A) = iff A is both open and closed. (iii) Bd( Int(A)) Bd(A). construct an example in which equality dos not hold. (iv) if A B = , then Bd(A) Bd(B) = Bd(A B ). Q12: Suppose X = Y = R and f : X Y is given by f (x) =1 if x 0 and f (x) = 1 if x 0. Describe the topology induced in X if Y has (a) the cofinite topology (b) the discrete topology (c) the usual topology (d) the L.R.T. MATH 373 PROBLEMS SHEET 3 Q2: Let A = { (x,y) R2 : 0 £ x £ 3, 1 £ y < 5 } R2. Describe Int(A), Bd(A), Ext(A), A, A , if R2 has the (i) usual topology (ii) cofinite topology (iii) L.R.T. (iv) indiscrete topology (v) discrete topology Q3: Let A and B be subsets of the spaces X and Y respectively. Prove that (i) A B A B (ii) Bd A B Bd ( A) B A Bd ( B) Q4: Let (X,T ) and (Y,T ) be topological spaces, and consider the product topology on X Y . Prove that A B X Y is dense in the space X Y iff A is dense in X and B is dense in Y. Q5: Let X denote the real line with the lower limit topology and let Y = X X . Prove that B = { [a,b) [c,d): a,b,c,d R, a < c, b < d } is a basis for Y. MATH 373 PROBLEMS SHEET 4 Q1: Let X= C [0,1] be the class of all continuous real valued functions on the closed interval [0,1], define d1 ,d 2 : X X R d by, for f , g X , 1 d1 f , g f ( x ) g ( x ) dx 0 d 2 f , g max f ( x ) g ( x ) Prove that d1 and d2 are metrics on X. Q2: Let X be the set of Riemann-integrable functions f from [0,1] into R. If d1 and d2 are defined as: 1 d1 f , g sup f ( x ) g ( x ) ; d 2 f , g f ( x ) g ( x ) dx 0 Show that d1 is a metric on X, but d2 is not a metric on X. (hint: only (M2) in the definition of a metric is not satisfied, to show that consider the function f (x) = 1 if x = 0 and f (x) = 0 if 0 < x 1.) Q3: Let (x,y) and (z,w) be any points in R2. Which of the following do not define metrics on R2? Justify your answer in each case. (i) d((x,y),(z,w)) = min x z , y w . (ii) d((x,y),(z,w)) = (x - z)2 + (y - w)2. (iii) d((x,y),(z,w)) x y z w . Q4: Let (X,d) be a metric space. Define on X X as follows: for (x,y) and (z,w) in X X, set ((x,y),(z,w)) = d(x,z) + d(y,w). Prove that is a metric on X X. Define a metric for X n. X n X X X (n - times). Q5: Show that neither of the functions given by: d x , y x y 2 ; ( x , y ) x y . 2 is a metric on R. Q6: Let R R , be the extended real line and let f : R 1,1 be a function defined by f x x if x R ; f 1 and f 1 1 x Show that the following function d x , y f ( x) f ( y ) defines a metric on R . Q7:Let Mn(R) be the set of real square matrix n n. If A=(aij) and B = (bij) are elements of Mn(R) and we define d on Mn(R) by d(A,B) = (i,j ( aij - bij )2 )1/2 Show that d is a metric on Mn(R). Q8: Let (X,d) be a metric space, let A be a subset of X. Show that the restriction of d to A A is a metric on A. Q9: (a) Let (X,d) be a metric space and A be a subset of X. Prove that x A iff d(x,A) = 0. (b)Show that a subset B of X is closed in X iff the subset {x X : d(x,B) = 0} B Q10: (a) Let (X,d) be a metric space, let p X. Prove that the subset A = { x X : d(x,p) } of X is closed in X. (b) Show that the closure of the open ball B(p, ) is not necessarily equal A. Q11: Give an example to show that, for an open -ball B(p, ) in a metric space (X,d), it is not necessarily true that Bd(B(p, )) = {x X: d(p,x) = }. Q12: Give an example of two closed, disjoint nonempty subsets A and B of a metric space where A B = and d(A,B) = 0. Q13: Let T be the upper limit topology on the real line R generated by the open - closed intervals (a,b]. Show that (R,T ) is Hausdorff. Q14: Let (X,T ) be a Hausdorff space. If T1 is a topology for X such that T1 T , then (X, T1) is also Hausdorff. Q15: Show that if a1 ,a2 , ,an are distinct points in a Hausdorff space X, then there exists disjoint open subsets U1 ,U 2 , ,U n of X such that a1 U1 ,a2 U 2 , ,an U n . Q16: Definition: “Let X be a non-empty set, the set x , x : x X is called the diagonal set”. Prove that a space X is Hausdorff iff the diagonal set is closed subset of X X. Q17: Let X be a finite set and d a metric for X. What is the metric topology for X Q18: Let (X,d) be a metric space . If is defined by the equation (x,y) = d(x,y)/(1+d(x,y)) Prove that is a bounded metric for X that generates the topologyT ( d ) of X. Q19: Show that in a metric space singleton elements are closed. Give an example to show that this does not hold for topological spaces in general. Q20: Let d be the function on Rn defined by d(x,y) = in1 xi yi Show that d is a metric on Rn induces the usual topology on Rn. Sketch the basis element under d when n = 2. Q21: For x = (-3,2) and y = (4,5) in R2, compute the distance between x and y under the following metrics:(i)usual, (ii) square (iii) discrete (iv) the metric in Q3. Q22: Determine the distance from the point(3,4) to the unit square [0,1] [0,1] with respect to the metrics in Q21. Q23:Calculate the diameter of the unit closed ball {(x,y,z) R3 : x2 + y2 + z2 1} with respect to the metrics in Q21. Q24: Show that a finite subset of a metric space has no limit points and is therefore a closed set. MATH 373 PROBLEMS SHEET 6 Q1: In R n , let Q denote the set of points having only rational coordinates and P its complement, the set of points having at least one irrational coordinate. Prove (i) Int(Q) = Int(P) = (ii) Q = P = R n , where Q , P are the derived sets. (iii) Bd(Q) = Bd(P) = R n . Q2: Let X be a space with the discrete metric. Show that every subset of X has empty boundary. Q3: Let X 1, d1 ,, X n , d n be metric spaces and let x x1,, xn and n y y1,, yn be arbitrary points in the product set X X .Show that i i 1 R defined by the function d : X X d ( x, y ) (d1( x1, y1 )) 2 (d n ( xn , yn )) 2 is a metric on the product set X . Q4: “ Definition: Let X , d be a metric space. A sequence xn in X is called Cauchy sequence iff for every 0 , there is a positive integer N s.t. d xn , xm whenever n,m>N.” Prove the following: (a) Every convergent sequence in X is a Cauchy sequence. (b) Give an example of a sequence in a metric space which Cauchy but does not converge in that space. Q5: “ Definition: A metric space X , d is complete if every Cauchy sequence xn in X converge to a point in X .” Prove the following: (a) A closed subspace A,d A of a complete metric space X , d is complete. (b) A metric space X is complete if every Cauchy sequence in X has a convergent subsequence. (c) The Euclidean space R n is complete in either of its usual metric or the square metric. MATH 373 PROBLEMS SHEET 7 Q1: Let Z be the integers as a subspace of R. Let f : Z Z be given by f x 2 x . Prove or disprove that f is continuous. Q2: Give an example of a bijective function f : X Y such that f is continuous, but f 1 is not continuous. Q3: Give an example of a function f : X Y and a subset A X such that f A is continuous but f is not continuous at any point of A. Q4: Prove or disprove: (i) All constant functions are continuous. (ii) If the domain of the function has the discrete topology, then the function is continuous. (iii) All injective (1-1) functions are continuous. (iv) If the range of the function has the indiscrete topology, then the function is cont. Q5: Let f : ( X ,T ) (Y ,T * ) , and let S be a subbase for T * . Show that f is continuous if and only if the inverse of every element of S is an open set in X. Q6: Let f : X 0,1 be a function from a topological space X into the unit interval 0,1 . Prove that if f 1a ,1 and f 10,b are open subsets of X for all 0 a ,b 1 , then f is continuous. Q7: Let f : X Y be continuous. If A X has a as a limit point, prove that f (a) is either a limit point of A or belongs to f (A). Give an example to show that if a is a limit point of A, then f (a) need not be a limit point of f (A). Q8: Let X = Y = R. define the function f : X Y if x 1 x f ( x) = x + 2 if x > 1 (i) Show that if X and Y have the usual topology, then f is not continuous. (ii) Show that if X and Y have the upper limit topology, then f is continuous. Q9: Suppose a function f : (X,T1) (Y,T2) is not continuous . Show that if T*1 is a topology on X coarser than T1 and T*2 is a topology on Y finer than T2, then f is also not continuous w.r.to T*1 and T*2. Q10: Show that the identity function I : (X, T ) (X,T* ) is cont. iff T is finer than T*. Q11: Let T : be a collection of topologies on a non empty set X . Prove that if a function f : X Y is continuous with respect to each T , then f is continuous with respect to the intersection topology T . Q12: Prove that the function f : 0,1 0,1 S1 0,1 defined by f ( x , y ) = (cos x ,sin x), y is continuous. Q13: Under what conditions a function f : X Y will not be continuous at a point x X ? Q14: Consider the topology defined on X a ,b,c ,d T X , ,a , b , a ,b , b,c ,d Let the function f : X X be defined as a b ,b d ,c b ,d c . Show that (i) f is not continuous at c. (ii) f is continuous at d. Q15: Suppose a singleton set ais an open subset of a topological space X. Prove that for any space Y and any function f : X Y , f is continuous at a X . Q16: Find an example of each of the following: (i) A close cont. function that is not open. (ii) A closed function that is not cont. (iii) An open function that is not closed. (iv) A cont. function that is neither closed nor open. (v) An open function that is not cont. Q17: Let the real function f : R R be defined by f x x 2 . Show that f is not open function. Q18: Let B be a base for a topological space X . Show that if f : X Y if f B is open in Y for every B B , then f is an open function. Q19: Let f : (X, T ) (X,T* ) be open and onto, and let B be a base for T . Is the collection {f [B]: B B } is necessarily a base for T* ? Q20: Prove that homeomorphism is an equivalence relation. Q21: Prove that a bijective function f : X Y is a homeomorphism if: (a) f is cont. and closed function. (b) f is cont. and an open function. (c) f and f -1 are both closed functions. (d) f and f -1 are both open functions. Q22: Let f : X Y be a 1-1 function from a topological space X onto the topological space Y. Show that the following are equivalent: (i) f is an open function. (ii) f -1 is cont. (iii) f is a closed function. Q23: Definition: “ Let (X,d) and (Y,) be two metric spaces. A function f : X Y is called isometry if f is onto and preserves distances i.e. d x1 , x2 f x1 , f x2 If there is an isometry between X and Y, then they said to be isometric” . Prove the following: (a) If f is an isometry, then f is 1-1.Moreover f and f -1 are cont. and thus a homeomorphism. Give an example to show that a homeomorphism need not be an isometry. (b) Isometry between metric spaces is an equivalence relation. Q24: Show that the unit n-sphere S n R n 1 is closed in R n 1 . (Hint: define a cont. function from R n 1 into R and show that S n is the inverse image of a closed set in. ) Q25: Show that the square whose vertices are the points (1,0), (0,1), (-1,0) and (0,-1) is homeomorphic to the unit circle S1 in R 2 . MATH 373 PROBLEMS SHEET 8 Q1: Let X be an infinite set with the cofinite topology. Prove that X is compact. Q2: Prove that the union of a finite number of compact sets is compact. Q3: (a) If (X,T ) is compact and T* T, then (X,T*) is compact. (b) Give an example to show that if T T*, then (X,T*) need not be compact. Q4: Show that any infinite subset of a discrete topological space is not compact. Q5: Let X and Y be compact Hausdorff spaces. Prove that f : X Y is cont. iff for each compact subset M of Y, f -1 (M) is compact in X. Q6: Show that if (X,T ) is compact and (X,T* ) is Hausdorff and if T* T, then T* = T. Q7: Prove that if A is compact and B is closed in a topological space X , then A B is compact. Q8: Let p: X Y Y be the projection. If X is compact and Y is an arbitrary space, prove that p is a closed function. Q9: Which of the following subsets of R is compact: (a) The natural numbers. (b) The rational numbers. Q10: Give an example of the following: (a) A closed subset of R3 which is not compact. (b) A bounded subset of R3 which is not compact. (c) A family of compact subsets of R whose union is not compact. Q11: Let A and B be open subsets of Rn , Prove that A B is compact iff A B . Q12: (a) Prove that any compact subset of a metric space is bounded and closed. (b) Give an example to show that a closed and bounded subset of a metric space need not be compact. (c) Prove that if A is a compact subset of a metric space X. Then the derived set A' is compact. Q13: (a) Show that a closed subset of a sequentially compact space is sequentially compact. (b) Show that a continuous image of a sequentially compact space is sequentially compact. Q14: Show that: (i) a continuous image of a limit point compact space need not be limit point compact. (b) a closed subset of a limit point compact space is limit point compact. Q15: Prove: Let f : X Y be a continuous function from a compact metric space X into a metric space Y. Then f (X ) is bounded subset of Y. Q16: Show that the unit n-sphere S n R n 1 is compact in R n 1 .