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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) =
in1 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 1a ,1 and f 10,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 ais 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 .