US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
Practical 3.1 : Examples of Metric Spaces, Normed Linear Spaces.
Objective Questions 3.1
(1) Consider the following maps d : R× R −→ R.
(i) d(x, y) = |x− 2y| (ii) d(x, y) = |x2 − y2|
(iii) d(x, y) = |x− y|2 (iv) d(x, y) = |x− y| 12
(a) (iii) and (iv) are metrics on R
(b) Only (iv) is a metric on R
(c) (ii) and (iii) are metrics on R
(d) (ii), (iii) and (iv) are metrics on R
(2) Let d1 and d2 be metrics on a non-empty set X. Then
(a) d21 + d
2
2, ad1 where a > 0 are metrics on X, where (d
2
1 + d
2
2)(x, y) =
(
d1(x, y)
)2
+(
d2(x, y)
)2
and (ad1)(x, y) = a
(
d1(x, y)
)
(b)
√
d1 +
√
d2, ad1 where a > 0 are metrics on X, where (
√
d1 +
√
d2)(x, y) =
√
d1(x, y) +√
d2(x, y) and (ad1)(x, y) = a
(
d1(x, y)
)
(c) ad1+bd2 where a, b ∈ R is a metric on X, where (ad1+bd2)(x, y) = ad1(x, y)+bd2(x, y)
(d) None of the above
(3) Consider the discrete metric d1 defined on a non-empty set X by d1(x, y) =
{
1 if x 6= y
0 if x = y
.
Then for x, y, z ∈ X,
(a) d1(x, z) < d1(x, y) + d1(y, z)
(b) d1(x, z) < d1(x, y) + d1(y, z) if and only if x, y, z are distinct.
(c) d1(x, z) = d1(x, y) + d1(y, z) if and only if x = y = z
(d) None of the above
(4) Let d1 and d2 be metrics on a non-empty setX. For x, y ∈ X, let d(x, y) = min {d1(x, y), d2(x, y)}
and d′(x, y) = max {d1(x, y), d2(x, y)}. Then
(a) Both d, d′ are metrics on X. (b) d is a metirc on X, d′ is not.
(c) d′ is a metirc on X, d is not. (d) None of the above.
(5) Let (X, d1) and (Y, d2) be metric spaces. d, d
′, d′′ : (X × Y ) × (X × Y ) −→ R are defined
as follows:
(i) d((x1, y1), (x2, y2)) = d1(x1, x2) + d2(y1, y2)
(ii) d′((x1, y1), (x2, y2)) = [(d1(x1, x2))2 + (d2(y1, y2))2]
1
2
(iii) d′′((x1, y1), (x2, y2)) = [(d1(x1, x2))2 + (d2(y1, y2))2]
(a) d, d′, d′′ are all metrics on X × Y (b) d, d′ are metrics on X × Y
(c) d′, d′′ are metrics on X × Y (d) None of the above.
(6) Let (X, ‖ ‖) be a normed linear space and x, y, z ∈ X. If d is the metric induced by the
norm then
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US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
(a) d(x+ z, y + z) ≥ d(x, y) and the strict inequality may hold.
(b) d(x+ z, y + z) ≥ d(x, y) + d(y, z) and the strict inequality may hold.
(c) d(x+ z, y + z) = d(x, y).
(d) None of the above
(7) Consider the norms ‖ ‖1, ‖ ‖2 and ‖ ‖∞ on R2, ‖x‖1 = |x1|+ |x2|, ‖x‖2 =
√
x21 + x
2
2, ‖x‖∞ =
max {|x1|, |x2|}. Then
(a) 2‖x‖∞ ≤ ‖x‖2 ≤ 2‖x‖1 (b) ‖x‖∞ ≤ ‖x‖2 ≤ ‖x‖1
(c) 2‖x‖∞ ≤ ‖x‖1 ≤ 2‖x‖2 (d) None of the above
(8) Let X = C[0, 1] and consider the norms ‖ ‖1, ‖ ‖∞ on X, where ‖f‖1 =
∫ 1
0
|f(t)| dt, ‖f‖∞ =
sup {|f(t)|, t ∈ [0, 1]}. Then for f(t) = t, g(t) = t2 ∈ X, if d1 and d∞ are metric induced
by ‖ ‖1, and ‖ ‖∞ then
(a) d1(f, g) =
1
2
, d∞(f, g) =
1
3
(b) d1(f, g) =
1
6
, d∞(f, g) =
1
4
(c) d1(f, g) =
1
3
, d∞(f, g) =
1
2
(d) None of the above.
(9) Consider the normed linear space (l2 , ‖ ‖2 ) where l2 = {(xn) : (xn) is a sequence over R, such that
∞∑
n=1
x 2n < ∞} and for x = (x1, x2, . . . , xn, . . .), ‖x‖2 =
√ ∞∑
n=1
x2n. Let e1 = (1, 0, 0, . . .), e2 =
(0, 1, 0, 0, . . .). Then for the metric d2 induced by ‖‖2,
(a) d2(e1 + e2, e1 − e2) =
√
2 (b) d2(e1 + e2, e1 − e2) = 2
(c) d2(e1 + e2, e1 − e2) = 1√
2
(d) None of the above.
(10) Let X be the set of all real sequences x = (xn). Consider the metric d defined by
d(x, y) = 0 if x = y
=
1
min {i : xi 6= yi} if x 6= y
where x = (xn), y = (yn) ∈ X. Then for distinct sequences x, y, z ∈ X
(a) d(x, z) ≤ d(x, y) + d(y, z) and the equality may hold.
(b) d(x, z) ≤ max {d(x, y), d(y, z)}
(c) d(x, z) ≥ max {d(x, y), d(y, z)}
(d) None of the above.
(11) Let (X, ‖ ‖) be a normed linear space and d be the metric induced by ‖ ‖. Then for
x, y, z ∈ X, d(x, z) = d(x, y) + d(y, z) if and only if
(a) y = z
(b) y lies on the segment joining x and z and between them.
(c) z lies on the segment joining x and y and between them.
(d) None of the above.
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US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
(12) Let X be a normed linear space and x, y ∈ X. Then
(a) ‖x− y‖ ≤ | ‖x‖ − ‖y‖ | (b) ‖x− y‖ = | ‖x‖ − ‖y‖ |
(c) ‖x− y‖ ≥ | ‖x‖ − ‖y‖ | (d) None of the above.
(13) Let X = M2(R). Consider the following maps from X −→ R.
(i) ‖A‖ = | det A|
(ii) ‖A‖ =
∑
1≤i,j≤2
|aij| where A = (aij)
(iii) ‖A‖ = max 1≤i,j≤2|aij| where A = (aij)
Then
(a) (i), (ii), (iii) are all norms on X.
(b) (ii) and (iii) are norms on X.
(c) (i) and (ii) are norms on X.
(d) None of the above.
Topology of Metric Spaces: Practical 3.1
Examples of Metric Spaces, Normed Linear Spaces
Descriptive Questions 3.1
(1) Let d1 and d2 be metrics on a non-empty set X. Check if the following are metrics on X.
Justify your answer.
(i) d, where d(x, y) = max {d1(x, y), d2(x, y)} for x, y ∈ X
(ii) d, where d(x, y) = min {d1(x, y), d2(x, y)} for x, y ∈ X
(iii) d, where d(x, y) = 2d1(x, y) + 3d2(x, y) for x, y ∈ X
(iv) d, where d(x, y) = (d1(x, y))
2 + (d2(x, y))
2 for x, y ∈ X
(v) d, where d(x, y) = max {1, d1(x, y), d2(x, y)} for x, y ∈ X
(2) Let (X, d) be a metric space. Show that the following are metrics on X.
(i) d1 where d(x, y) =
√
d(x, y)
(ii) d, where d(x, y) =
d(x, y)
1 + d(x, y)
(3) Show that d is a metric on R, where d(x, y) =
{
0 if x = y
|x|+ |y| if x 6= y, x, y ∈ R
(4) Let Rn = {(x1, x2, . . . , xn) : xi ∈ R for 1 ≤ i ≤ n}. Show that ‖ ‖1, ‖ ‖2, and ‖ ‖∞
are norms on Rn where for x = (x1, x2, . . . , xn), ‖x‖1 =
i=n∑
i=1
|xi|, ‖x‖2 =
√√√√ i=n∑
i=1
x2i and
‖x‖∞ = max {|x1| : 1 ≤ i ≤ n}. Further, show that ‖x‖∞ ≤ ‖x‖2 ≤ ‖x‖1 and
‖x‖1 ≤
√
n‖x‖2 ≤ n‖x‖∞ for x ∈ Rn
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US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
(5) Let l2 = {(xn) : (xn) is a sequence of real numbers such that
∞∑
n=1
x2n < ∞}. If ‖x‖2 =( ∞∑
n=1
x2n
) 1
2
for x = (x1, x2, . . . , xn, . . .) ∈ l2 , then show theat (l2 , ‖ ‖2 ) is a normed linear
space.
(6) Let X = C [0 , 1 ] and show that ‖ ‖1 : X −→ R and ‖ ‖∞ : X −→ R defined by,
‖f‖1 =
∫ 1
0
|f(t)| dt, ‖f‖∞ = sup {|f(t)| : t ∈ [0, 1]} are norms on X
(7) Let X = C [0 , 1 ] and consider the norms ‖ ‖1 and ‖ ‖∞ defined by,
‖f‖1 =
∫ 1
0
|f(t)| dt, ‖f‖∞ = sup {|f(t)| : t ∈ [0, 1]}
Then for f = t, g = t2, h = t3, t ∈ [0 , 1 ], find d1(f, g), d∞(f, g), d1(f, h), d∞(f, h) where d1
and d∞ are metrics induced by the norms ‖ ‖1 and ‖ ‖∞ respectively.
(8) Let X be the set of real sequences
(i) Show that d : X ×X −→ R defined by
d(x, y) = 0 if x = y
=
1
min {i : xi 6= yi} if x 6= y
where x = (xn), y = (yn) ∈ X is a metric on X.
(ii) Show that d : X ×X −→ R defined by
d(x, y) =
∞∑
i=1
|xi − yi|
2i(1 + |xi − yi|)
where x = (xn), y = (yn) ∈ X is a metric on X.
(iii) Let X = {(xn) : (xn) is a sequence of real numbers, xn −→ 0}. Show that ‖ ‖ : X −→
R defined by ‖x‖ = sup {|xn| : n ∈ N} for x = (xn) is a norm on X.
(9) Let ‖ ‖2 be the Euclidean norm on R2. Let d : R2 × R2 −→ R be defined by
d(x, y) = ‖x‖2 + ‖y‖2 if x 6= y
= 0 if x = y
for x, y ∈ R2. Show that d is a metric on R2
(10) Show that d is a metric on N where for m,n ∈ N,
d(m,n) = 0 if m = n
= 1 +
1
m+ n
if m 6= n
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US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
(11) Show that ‖ ‖ is a norm on X, where X = M2(R) and ‖A‖ = max 1≤i,j≤2|aij| for A = (aij)
(12) Show that ‖ ‖1 is a norm on l1 where l1 =
{
(xn) : xn ∈ R,
∞∑
n=1
|xn| < ∞
}
and ‖x‖1 =
∞∑
n=1
|xn| for x = (xn)
(13) Show that C (set of complex numbers) is a normed linear space where norm is the absolute
value of a complex number.
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US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
Topology of Metric Spaces Practical 3.2
Sketching of Open Balls in R2, Open and Closed sets, Equivalent metric spaces
Objective Questions 3.2
. (Revised Syllabus 2018-19)
(1) In a metric space (X, d)
(a) an arbitrary intersection of open sets is an open sets.
(b) an arbitrary intersection of open balls is an open ball.
(c) an intersection of finitely many open balls is an open ball.
(d) None of the above.
(2) Let (X, d) be a metric space and x, y ∈ X, r, s > 0. If B(x, r) = B(y, s), then
(a) x = y and r = s (b) x = y but r may not be equal to s
(c) r = s (d) None of the above
(3) Let (X, d) be a metric space and x, y ∈ X, 0 < r < s. Then
(a) B(x, r) ⊆ B(x, s) and the equality may occur. (b) B(x, r) ( B(x, s),
(c) B(x, r) = B(x, s) if r ≥ 1 (d) None of the above.
(4) Let (X, d) be a metric space in which the only open subsets are ∅ and X. Then
(a) d is a discrete metric on X.
(b) For x, y ∈ X, d(x, y) ≥ 1 if x 6= y
(c) X is a singleton set.
(d) None of the above.
(5) Let G be a non-empty bounded open set in R2 with Euclidean metric. Then G is of the
type
(a) (a, b)× (c, d), where a, b, c, d ∈ R, a < b, c < d.
(b) I × J, where I and J are union of finitely many bounded open intervals in R
(c) G1 ×G2, where G1 and G2 are bounded open subsets of R.
(d) None of the above.
(6) Consider the normed linear space (R2, ‖ ‖1) where for x = (x1, x2) ∈ R2, ‖x‖1 = |x1|+ |x2|.
If B1((0, 0), 1) is an open ball with center (0, 0) and radius 1, then
(a) B1((0, 0), 1) is a square with sides of length
√
2 which are parallel to coordinate axes.
(b) B1((0, 0), 1) is a square with sides of length
√
2 and diagonals are parallel to
coordinate axes.
(c) B1((0, 0), 1) is a square with sides of length 2 which are parallel to coordinate axes.
(d) None of the above.
(7) Let (X, d) be a metric space and x, y ∈ X. Let d(x, y) = s > 0. Then B(x, r)∩B(y, r) = ∅,
if
(a) r ≥ s
2
(b) 0 < r ≤ s
2
(c) r ≥ 2s (d) None of the above
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US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
(8) Consider the normed linear spaces (R2, ‖ ‖1), (R2, ‖ ‖2) and (R2, ‖ ‖∞) where for x =
(x1, x2) ∈ R2 ‖x‖1 = |x1|+ |x2|, ‖x‖2 =
√
x21 + x
2
2, ‖x‖∞ = max {|x1|, |x2|}
If B1((0, 0), 1), B2((0, 0), 1) and B∞((0, 0), 1) denote open balls in (R2, ‖ ‖1), (R2, ‖ ‖2) and
(R2, ‖ ‖∞) respectively. Then
(a) B1((0, 0), 1) ( B2((0, 0), 1) ( B∞((0, 0), 1)
(b) B1((0, 0), 1) = B2((0, 0), 1) = B∞((0, 0), 1)
(c) B∞((0, 0), 1) ( B2((0, 0), 1) ( B1((0, 0), 1)
(d) None of the above.
(9) Let (X, d) be a metric space aned d1 be the metric on X defined by d1(x, y) =
d(x, y)
1 + d(x, y)
for x, y ∈ X
(a) Every open ball in (X, d1) is an open ball in (X, d) and viceversa.
(b) Every open ball in (X, d1) except possibly B(x, r), r ≥ 1 for any x ∈ X is an
open ball in (X, d) .
(c) Every open ball in (X, d1) is an open ball in (X, d)
(d) None of the above.
(10) Let (X, ‖ ‖) be a normed linear space. Let A ⊆ X and U be an open subset of X in (X, d)
where d is the metric induced by ‖ ‖. Then
(a) A+ U is open if and only if A is open.
(b) A+ U is open.
(c) A+ U is open if and only if A = ∅ or A is a singleton set.
(d) None of the above.
(11) Let (X, d) be a metric space , a ∈ X and r′ > r > 0. Let U1 = {x ∈ X : d(x , a) >
r},U2 = {x ∈ X : d(x , a) 6= r} and U3 = {x ∈ X : r < d(x , a) < r ′}. Then
(a) U1 and U2 are open subsets of X, but U3 may not be open.
(b) U1 ,U2 ,U3 are all open.
(c) U1 is open subset of X, but U2 and U3 may not be open.
(d) None of the above.
(12) Consider the metric spaces (N, d) and (N, d1) where d is the usual distance (induced from
R) and d1 is the discrete metric in N. Then
(a) d and d1 are equivalent metrics on N, but the two metric spaces do not have
same open balls.
(b) The open balls in two metric spaces are the same.
(c) Every open ball in (N, d) is an open ball in (N, d1)
(d) None of the above.
(13) Consider the following subsets of C with respect to the usual distance
(i) A = {z ∈ C : z = 2}⋃{z ∈ C : |z| < 2}
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US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
(ii) B = {z ∈ C : |Re z| < a} where a > 0, a ∈ R
(iii) C = {z ∈ C : z 6= i
n
, n ∈ N}
(a) A,B and C are open.
(b) B,C are open.
(c) Only B is open.
(d) Only C is open.
(14) Consider the following subsets (R3, d) where d Euclidean.
E = {(x, y, 0) ∈ R3}
F = {(x, y, z) ∈ R3 : ax+ by + cz = d, at least one of a, b, c is not zero}
G = {(x, y, z) ∈ R3 : xyz 6= 0}. Then
(a) E,F and G are not open.
(b) Only G is open.
(c) F,G are open.
(d) Only E is open.
(15) Let X = C [0 , 1 ] with norm ‖ ‖∞.
Let E = {f ∈ X : f(0) 6= 0}, F = {f ∈ X : f(1
2
) 6= 0}. Then
(a) E is not open and F is open.
(b) Neither E nor F are open.
(c) Both E and F are open.
(d) E is open but F is not.
(16) Let X = C [0, 1]. Then
(a) B1(0, 1) is open in (X, ‖ ‖∞)
(b) B1(0, 1) ⊆ B∞(0, r) for some r > 0.
(c) B∞(0, 1) ⊆ B1(0, r) for some r > 0.
(d) None of the above.
Topology of Metric Spaces: Practical 3.2
Sketching of Open Balls in R2, Open and Closed sets, Equivalent metric spaces
Descriptive Questions 3.2
(1) Give an example of a metric space in which B(x, r) = B(y, s) but x 6= y and r 6= s.
(2) Determine which of the following sets are open in the given metric space. Justify your
answer in each case.
(i) U = {(x , y) ∈ R2 : xy 6= 0} with Euclidean metric.
(ii) U = {(x , y) ∈ R2 : x = 0} with Euclidean metric.
(iii) Q in R with usual distance.
(iv) U = {(x , y) ∈ R2 : x 2 − y2 ≤ 1} with Euclidean metric.
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US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
(v) U = {(x , y) ∈ R2 : 2x + 3y < 1} with Euclidean metric.
(ii) U = B((0 , 0 ), 1 ) \ {(1
2
, 1
2
), (1
3
, 1
3
)} ∈ R2 with Euclidean metric.
(3) Let (X, d) be a discrete metric space and x ∈ X. Find
(i) B(x, 1
2
) (ii) B(x, 3
4
) (iii) B(x, 1) (iv) B(x, r), r > 1
(4) Draw open ball B((0, 0), 1) in R2 with respect to the given metric.
(i) d1 induced by the norm ‖ ‖1, ‖x‖1 = |x1|+ |x2| for x = (x1, x2) ∈ R2
(ii) d2, the Euclidean metric.
(iii) d1 induced by the norm ‖ ‖∞, ‖x‖∞ = max {|x1|, |x2|} for x = (x1, x2) ∈ R2
(iv) d where d(x, y) = 2|x1 − y1|+ 3|x2 − y2| for x = (x1, x2), y = (y1, y2) ∈ R2
(5) Show that in the following examples U is open subset of (R2, d), where d is the Euclidean
metric. Also, for p ∈ U , find maximum rp such that B(p, rp) ⊆ U .
(i) U = {(x , y) ∈ R2 : x > 0 , y > 0} .
(ii) U = {(x , y) ∈ R2 : x /∈ Z, y /∈ Z} .
(iii) U = (0 , 1 )× (0 , 1 ) .
(iv) U = {(x , y) ∈ R2 : −1 < x + y < 1} .
(6) Let f, g ∈ C [0 , 1 ] and suppose f(t) < g(t) for each t ∈ [0, 1]. Show that U = {h ∈ C [0 , 1 ] :
f (t) < h(t) < g(t) for each t ∈ [0 , 1 ]} is an open subset of X = C [0 , 1 ] under ‖ ‖∞ norm
where ‖f‖∞ = sup {|f(t)| : t ∈ [0, 1]}
(7) Consider X = C [0 , 1 ] under the norms ‖ ‖1 and ‖ ‖∞ where ‖f‖1 =
∫ 1
0
|f(t)| dt and
‖f‖∞ = sup {|f(t)| : t ∈ [0, 1]}. Draw the open ball B(0, 1) in (X, ‖ ‖1) and (X, ‖ ‖∞).
(meaning show when does f ∈ C [0 , 1 ] lie in the open ball B(0, 1)).
(8) Describe the open balls B(p, r) for p ∈ Z, r > 0 considering cases 0 < r < 1, r = 1, r > 1 in
the subspace Z of R with usual distance.
(9) Let (X, d1) and (Y, d2) be metric spaces. Consider the metric d : (X×Y )× (X×Y ) −→ R
defined by d((x1, y1), (x2, y2)) = max {d1(x1, x2), d2(y1, y2)}. Let p ∈ X, q ∈ Y and r, s > 0.
Show that B(p, r)×B(q, s) is an open set in (X × Y, d).
(10) Consider the metric δ on R2 defined by
δ(x, y) = ‖x‖+ ‖y‖ if x 6= y
= 0 if x = y
for x, y ∈ R2 where ‖ ‖ is the Euclidean norm in R2. Find the open balls B((0, 0), r) and
B(x, r) where x 6= (0, 0), ‖x‖ =  and 0 <  < r
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US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
(11) Check whether the following subsets of C with respect to usual distance are open. Justify
your answer.
1. A = {z ∈ C : z = 2}⋃{z ∈ C : |z| < 2}
2. B = {z ∈ C : |Re z| < a, where a ∈ R+}
3. C = {z ∈ C : z 6= i
n
, n ∈ N}
(12) Let (X, d) be a metric space. We define a metric d′ on X ×X by
d′((x1, x2), (y1, y2)) = max {d(x1, y1), d(x2, y2)}
Show that D = {(x, x) : x ∈ X} is a closed subset of (X ×X, d′)
(13) Show that S ′ = {(x, y) ∈ R2 : x2 + y2 = 1} is a closed subset of (R2, ‖ ‖2), ‖ ‖2 being the
Euclidean metric.
(14) In the following examples, show that the given pairs of metrics are equivalent.
(i) For a metric space (X, d), the metrics d and d1, where d1(x, y) =
d(x, y)
1 + d(x, y)
, x, y ∈ X
(ii) For a metric space (X, d), the metrics d and d1, where d1(x, y) = min {1, d(x, y)}, x, y ∈
X
(iii) On N, d and d1 where d is the induced metric from the usual distance d in R and d1
is the discrete metric.
(15) Let X = C [0 , 1 ] and d1 and d∞ be the metrics on X induced by ‖ ‖1 and ‖ ‖∞. Prove or
disprove d1 and d∞ are equivalent metrics on X.
(16) Let d1, d2, d∞ be three metrics defined on R2 as follows:
d1(x, y) = |x1 − y1|+ |x2 − y2|, d2(x, y) =
√
(x1 − y1)2 + (x2 − y2)2
d∞(x, y) = max {|x1 − y1|, |x2 − y2|}, ∀x = (x1, x2) & y = (y1, y2).
Prove that d1, d2, d∞ are equivalent metrics on R2 by showing
d∞(x, y) ≤ d2(x, y) ≤
√
2d∞(x, y) and d∞(x, y) ≤ d1(x, y) ≤ 2d∞(x, y).
(17) Let d1, d2, d∞ be three metrics defined on Rn as follows:
d1(x, y) =
n∑
i=1
|xi − yi|, d2(x, y) =
√
n∑
i=1
(xi − yi)2
d∞(x, y) = max {|xi − yi| : 1 ≤ i ≤ n}
∀x = (x1, x2, . . . , xn) & y = (y1, y2, . . . , yn)
Show that d1(x, y) ≥ d2(x, y) ≥ d∞(x, y) ≥ n− 12d2(x, y) ≥ n−1d1(x, y)
10
US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
Topology of Metric Spaces: Practical 3.3
Subspaces, Interior points, Limit Points, Dense Sets and Separability, Diameter of
a set, closure
Objective Questions 3.3
. (Revised Syllabus 2018-19)
(1) Consider the subspace Z of the metric subspace R with usual distance. Then
(a) Every open ball in Z is an infinite set.
(b) Every open ball in Z is a singleton set.
(c) Every open ball in Z is a finite set.
(d) None of the above.
(2) Let (X, d) be a metric space and A,B ⊆ X. Then
(a) (A∪B)◦ = A◦∪B◦, (A∩B)◦ = A◦∩B◦ (b) (A∪B)◦ ⊆ A◦∪B◦, (A∩B)◦ ⊆ A◦∩B◦
(c) A◦ ∪B◦ ⊆ (A ∪B)◦, (A ∩B)◦ = A◦ ∩B◦ (d) None of the above.
(3) Let A be a non-empty subset of R, (distance being usual) then A◦ can be
(a) empty
(b) singleton set
(c) a finite set containing more than one element
(d) countable but not finite
(4) Consider A = [0, 1) with the induced distance from the usual distance in R. Then
(a) An open ball in A is of the type (−r, r) with 0 < r < 1
(b) [0, 1
2
) is an open ball in A
(c) [0, 1) is not an open ball in A
(d) None of the above
(5) In the subspace (Q, d) of (R, d) where d is the usual distance in R, E = {r ∈ Q : 2 < r2 < 3}
is
(a) an open ball (b) an open set which is not bounded.
(c) open and closed (d) None of the above.
(6) Let A be a closed subset of R (distance usual) A 6= ∅, A 6= R. Then
(a) A = (A◦)
(b) A is countable.
(c) A is not open.
(d) A is a bounded set.
(7) Let (X, d) be a metric space and A,B ⊆ X. Let D(S) denote the set of limit points of
S ⊆ X. Then
(a) If A ( B, then D(A) ( D(B)
(b) If A ( B, then D(B) ( D(A)
(c) If A ( B, then D(A) ⊆ D(B) and the equality may occur.
(d) None of the above.
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US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
(8) Let d be the usual distance on R and d1 be the discrete metric on R. Let A = (0, 1). If
D(A) denotes the set of all limit points of A, then
(a) In (R, d), D(A) = (0, 1) and in (R, d1), D(A) = {0, 1}
(b) In (R, d), D(A) = [0, 1] and in (R, d1), D(A) = ∅
(c) In (R, d), D(A) = (0, 1) and in (R, d1), D(A) = (0, 1)
(d) None of the above.
(9) Consider the following subsets of R (distance in R being usual):
(i) N (ii) Q (iii) { 1
n
: n ∈ N} (iv) (−1, 0) . Then 0 is a limit point of
(a) (iv) only
(b) (ii), (iii) and (iv)
(c) (ii) and (iv) only
(d) N
(10) Let (X, d) be a metric space and A,B ⊆ X. Then
(a) A ∪B = A ∪B,A ∩B = A ∩B
(b) A ∪B ⊂ A ∪B,A ∩B = A ∩B
(c) A ∪B = A ∪B,A ∩B ⊆ A ∩B
(d) None of the above
(11) Let (X, d) be a metric space and A ⊆ X. If G ⊆ X is an open set such that G ∩ A = ∅
then
(a) G ∩ A = ∅ (b) G ∩ A = ∅ (c) G ∩ A = ∅ (d) None of the above
(12) Let A = {1, 1
2
, 1
3
, 2
3
, 1
4
, 3
4
, 1
5
, 2
5
, 3
5
, 4
5
, · · · } in R where the distance is usual. Then
(a) A is a closed set. (b) A is not a closed set, A = (0, 1]
(c) A is not a closed set, A = [0, 1]
(d) None of the above.
(13) Consider Y = [0, 1] ⊆ R, with the induced usual distance d of R. Let A = [0, 1) ⊆ Y. Then
in (Y, d)
(a) ∂A = (0, 1) (b) ∂A = {0, 1} (c) ∂A = {1} (d) None of the above.
(14) Consider N with the induced usual distance of R. Let A = {1, 2, . . . , 10} ⊆ N. Then the
statement which is not true in (N, d) is
(a) A◦ = ∅ (b) A = A (c) ∂A = ∅ (d) None of the above.
(15) Let A,B ⊂ R, and d be the usual distance in R. Then
(a) d(A◦, B◦) = d(A,B) = d(A,B) (b) d(A,B) = d(A,B)
(c) d(A◦, B◦) = d(A,B). (d) None of the above.
(16) Let (X, d) be a metric space and A,B ⊆ X such that A,B are non-empty and A∩B = ∅.
Then
(a) d(A,B) > 0 (b) d(A,B) > 0 if A,B are open.
(c) d(A,B) > 0 if A,B are closed. (d) None of the above.
12
US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
(17) Let S1 = {(x, y) : x2 + y2 = 1} ⊆ R2, distance d being Euclidean. For p ∈ R2, d(p, S1)
equals
(a) ‖p‖ (b) ‖p‖ − 1 (c) ‖p‖+ 1 (d) None of the above.
(18) Let A = {1, 1
2
, 1
4
, 3
4
, 1
8
, 3
8
, 5
8
, 7
8
, · · · } (distance in R usual). Then A equals
(a) [0, 1] (b) (0, 1) (c) [0, 1] ∩Q (d) {m
2n
,m, n ∈ N} ∩ [0, 1]
(19) Consider the set A = {1, 1
2
, 1
3
, 2
3
, 1
4
, 3
4
, 1
5
, 2
5
, 3
5
, 4
5
, · · · } (distance in R usual). Then A equals
(a) A is a closed set (b) A is not a closed set, A = (0, 1].
(c) A is not a closed set, A = [0, 1] (d) None of the above.
(20) Let A =
{ |x|
1 + |x| : x ∈ R
}
, (distance usual). Then the set of all limit points of A is
(a) (0, 1] (b) (0,∞) (c) [0, 1] (d) None of the above.
(21) Let A =
{
x
1 + |x| : x ∈ R
}
, (distance usual). Then the set of all limit points of A is
(a) (−1, 1) (b) [−1, 1] (c) (0,∞) (d) None of the above.
Topology of Metric Spaces: Practical 3.3
Subspaces, Interior points, Limit Points, Dense Sets and Separability, Diameter of
a set, closure
Descriptive Questions 3.3
(1) Give an example of a metric space (X, d), A,B ⊆ X such that A◦ = B◦ = ∅ but (A∪B)◦ =
X
(2) Find the interiors of the following subsets in a given metric space.
(i) Z in (R, d) where d is the usual distance.
(ii) Q in (R, d) where d is the usual distance.
(iii) {(x, y) ∈ R2 : x > y} ∪ {(0, 0)} in (R2, d) where d is the Euclidean metric.
(3) Find the closure of the following subsets of C (distance being usual)
(i) S = {z = i
n
: n ∈ N}
(ii) S = {z = 1
m
+ i
n
: m,n ∈ N}
(iii) S = {z = x+ iy, x, y ∈ (0, 1), x, y ∈ Q}
(iv) S = {z = x+ iy, x, y ∈ (0, 1)}
(4) Consider the subspace A = [0, 1) of R where distance in R is usual. Find BA(0, r) an open
ball in the subspace A for r > 0
(5) Consider the subspace A = [0,∞) of R where distance in R is usual. Find BA(0, 1) an open
ball in the subspace (A, d).
13
US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
(6) Show that A = {x ∈ Q : −√2 < x < √2} is both open and closed in the subspace Q of R
with usual distance.
(7) Prove or disprove : Let (X, d) be a metric space and A ⊆ X. Then
(i) (A◦) = A (ii) (A)◦ = A◦
(8) In R, with respect to usual distance, show that A = N, B = {n + 1
n
: n ∈ N, n 6= 1} are
closed sets such that A ∩B = ∅. Also find d(A,B).
(9) (i) In (R, d), where d is the usual distance, find d(Q,R \Q) and d(Q, A) where A is any
non-empty subset of R.
(ii) In (R2, d), d being Euclidean, find d(A,B) where A = {(x, y) ∈ R2 : xy = 0} and
B = {(x, y) ∈ R2 : xy = 1}.
14
US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
Topology of Metric Spaces: Practical 3.4
Limit Points, Sequences, Bounded, Convergent and Cauchy Sequences in a
Metric Space
Objective Questions 3.4
. (Revised Syllabus 2018-19)
(1) Let (xn) be a sequence in a metric space (X, d), xn −→ p. Let A = {xn : n ∈ N}. Then
(a) p is a limit point of A
(b) p ∈ A
(c) There is a subsequence (xnk) of (xn) having distinct terms such that xnk −→ p
(d) None of the above.
(2) Let S be an infinite subset of R such that S ∩Q = ∅. Then
(a) S has a limit point which belongs to R \Q.
(b) S has a limit point which belongs to Q.
(c) S is not closed.
(d) R \ S has a limit point which is in S.
(3) Let d1 and d2 be equivalent metrics on X and (xn) be a sequence in X. Then
(a) (xn) is bounded in (X, d1)⇐⇒ (xn) is bounded in (X, d2).
(b) (xn) is convergent in (X, d1)⇐⇒ (xn) is convergent in (X, d2).
(c) (xn) is a Cauchy sequence in (X, d1) ⇐⇒ (xn) is a Cauchy sequence in (X, d2).
(d) None of the above.
(4) Every Cauchy sequence is eventually constant in
(a) (N, d) where d is usual.
(b) (Q, d) where d is usual.
(c) (R \Q, d) where d is usual.
(d) None of the above.
(5) d and d1 are metrics on X = (0,∞) where d is the usual distance and d1(x, y) =
∣∣∣∣1x − 1y
∣∣∣∣.
Then
(a) If (xn) is a Cauchy sequence in (X, d1) then (xn) is a Cauchy sequence in (X, d)
(b) If (xn) is a Cauchy sequence in (X, d) then (xn) is a Cauchy sequence in (X, d1)
(c) If (xn) is Cauchy in (X, d1), (xn) may not be Cauchy in (X, d).
(d) (xn) is a Cauchy sequence in (X, d)⇐⇒ (xn) is Cauchy sequence in (X, d1)
(6) d and d1 are metrics on X = (0,∞) where d is the usual distance and d1(x, y) =
∣∣∣∣1x − 1y
∣∣∣∣.
Then
(a) If (xn) is a bounded sequence in (X, d1) then (xn) is a bounded sequence in (X, d)
(b) If (xn) is a bounded sequence in (X, d) then (xn) is a bounded sequence in (X, d1)
(c) If (xn) is bounded in (X, d1), (xn) may not be bounded in (X, d).
(d) (xn) is a bounded sequence in (X, d)⇐⇒ (xn) is bounded sequence in (X, d1)
15
US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
(7) Let d1 and d2 be metrics on X such that k1d2(x, y) ≤ d1(x, y) ≤ k2d2(x, y) for all x, y ∈ X
where k1, k2 > 0 are constants. The the statement which is not true is
(a) (xn) is Cauchy in (X, d1) if and only if (xn) is Cauchy in (X, d2).
(b) xn −→ p in (X, d1) if and only if xn −→ p in (X, d2).
(c) (xn) is bounded in (X, d1) if and only if (xn) is bounded in (X, d2).
(d) None of the above.
(8) Consider the sequence (xk) defined by xk =
(
(−1)k, 1
k
)
in R2. d and d1 are metrics on R2
where d is the Euclidean distance and d1 is discrete metric. Then
(a) (xk) is not bounded in (R2, d) and (R2, d1). (b) (xk) converges in (R2, d).
(c) (xk) has a convergent subsequence in (R2, d). (d) (xk) converges in (R2, d1).
(9) Let xk −→ x and yk −→ y in (Rn, d), d is Euclidean distance. Which statement is not
true?
(a) ‖xk‖ −→ ‖x‖ and ‖yk‖ −→ ‖y‖.
(b) 〈xk, yk〉 −→ 〈x, y〉
(c) x is a limit point of the set A = {xk : k ∈ N} and y is a limit point of the set
B = {yk : k ∈ N}
(d) xk + yk −→ x+ y
(10) Consider X = C[0, 1], ‖f‖1 =
∫ 1
0
|f(t)| dt, ‖f‖∞ = sup {|f(t)| : t ∈ [0, 1]} ∀f ∈ X and
fn(x) = x
n. Then
(a) {fn} converges in (X, ‖ ‖1) but not in (X, ‖ ‖∞)
(b) {fn} converges in (X, ‖ ‖∞) but not in (X, ‖ ‖1)
(c) {fn} does not converge in both.
(d) {fn} converges in both.
(11) Consider (N, d) where d(m,n) =
{
0 if m = n
1 +
1
m+ n
if m 6= n Then
(a) Every sequence in (N, d) is bounded.
(b) Every sequence in (N, d) is eventually constant.
(c) Every Cauchy sequence in (N, d) is eventually constant.
(d) Every sequence in (N, d) is Cauchy.
(12) Consider the sequence xn = n− [
√
n] in (R, d) where d is usual metric. Then
(a) (xn) is Cauchy.
(b) (xn) is monotone increasing.
(c) (xn) is monotone decreasing.
(d) (xn) is not convergent but has a convergent subsequence.
16
US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
(13) Let d1 and d2 be two metrics on X and there exists real numbers k1, k2 > 0 such that
k1d2(x, y) ≤ d1(x, y) ≤ k2d2(x, y) ∀x, y ∈ X. Mark the sentences which is not true.
(a) (xn) is a Cauchy sequence in (X, d1) implies (xn) is a Cauchy sequence in (X, d2)
(b) (xn) is a bounded sequence in (X, d1) implies (xn) is a bounded sequence in (X, d2)
(c) (xn) is a convergent sequence in (X, d1) implies (xn) is a convergent sequence in (X, d2)
(d) (a), (b) and (c) are not true.
(14) The sequence
( 1
n
)
is not convergent in
(a) [0, 1] with usual distance.
(b) [0, 1] with discrete metric.
(c) Q with usual distance. (d) [0,∞) with usual distance.
(15) The Cauchy sequence which is convergent in (Q, d), where d is the usual distance, is
(a) (xn), where xn = 1 +
1
1!
+ +
1
2!
· · ·+ + 1
n!
(b) (xn) where x1 = 1 and xn =
1
2
(
xn +
2
xn
)
(c) (xn) = {0.1, 0.101, 0.101001, 0.1010010001, · · · } (d) (xn) where xn = 1
n
(
1 +
1
n
)n
Topology of Metric Spaces: Practical 3.4
Sequences, convergent and Cauchy sequences in a metric space
Descriptive Questions 3.4
(1) Show that the following sequences in R2 are convergent, distance being Euclidean.
(i) (xn) where xn =
(
1
n2
,
n2 − 1
n3 + 1
)
(ii) (xn), where xn =
(
2n,
1
n
)
for n ≤ 9 and xn =
(
210,
−1
n
)
for n ≥ 10
(2) Prove or disprove: Let d1, d2 be equivalent metrics on a non-empty set X. Then
(i) (xn) is bounded in (X, d1) if and only if (xn) is bounded in (X, d2)
(ii) (xn) is Cauchy in (X, d1) if and only if (xn) is Cauchy in (X, d2)
(3) Let d1 and d2 be equivalent metrics on a non-empty set X such that there exist k1, k2 > 0
such that
k1d1(x, y) ≤ d2(x, y) ≤ d2d1(x, y) ∀x, y ∈ X
Then show that
(i) (xn) is bounded in (X, d1) if and only if (xn) is bounded in (X, d2)
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US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
(ii) (xn) is Cauchy in (X, d1) if and only if (xn) is Cauchy in (X, d2)
(4) Show that the sequence xn =
1
n
converges to 0 in the usual metric space R but is not
convergent in X = (0, 1) with the usual metric.
(5) X = C [0 , 1 ]. Show that fn(t) = e
−nt converges to 0 w.r.t. the metric d1(f, g) =
∫ 1
0
|f(x)−
g(x)| dx but is not convergent w.r.t. the metric d∞(f, g) = sup{|f(x)− g(x)| : x ∈ [0, 1]}
(6) Let (X, d) be a metric space. If (xn) and (yn) are sequences in X such that xn −→ x and
yn −→ y, then prove that the sequence d(xn, yn) −→ d(x, y) in R w.r.t. the usual metric.
(7) Let X = C [0 , 1 ] be a metric space with the metric d∞ defined by
d∞(f, g) = sup{|f(t)− g(t)| : t ∈ [0, 1]}
Show that the sequence {fn} in X given by fn(t) = nt
n+ t
∀t ∈ [0, 1], is a Cauchy sequence
in X.
(8) Prove that every Cauchy sequence in a discrete metric space is convergent.
(9) Let (xn) be a Cauchy sequence in a metric space (X, d) and (xnk) be a subsequence of (xn).
Show that d(xn, xnk) −→ 0 in R w.r.t. the usual metric.
(10) Let (xn) and (yn) be Cauchy sequences in a metric space (X, d). Prove that (d(xn, yn)) is
a Cauchy sequence in R w.r.t. the usual distance.
(11) Let (X, d) be a metric space and d′ be a metric on X defined by
d′(x, y) = min{1, d(x, y)}
Show that (xn) is a Cauchy sequence in (X, d) if and only if it is a Cauchy sequence in
(X, d′).
(12) Let (X, d1) be a metric space and (xn) be a sequence in X. Show that xn −→ x in (X, d1)
if and only if d1(xn, x) −→ 0 in (R, d) where d is the usual distance in R.
(13) Let (an) and (bn) be sequences in a metric space (X, d1) and xn = d(an, bn).If (an) is a
Cauchy sequence in (X, d1) and xn −→ 0 in (R, d) (d is the usual distance), then show that
(bn) is a Cauchy sequence.
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US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
Topology of Metric Spaces: Practical 3.5
Complete Metric Spaces
Objective questions 3.5
. (Revised Syllabus 2018-19)
(1) Fn = [n,∞) for each n ∈ N. Then ∩n∈NFn
(a) has infinitely many points (b) is a singleton set.
(c) is the empty set. (d) None of the above .
(2) In R with respect to usual distance ∩n∈NFn is a singleton set when
(a) Fn = [−n, n] (b) Fn = [n, n+ 1] (c) Fn = [1− 1n , 1] (d) Fn = [0, n]
(3)
⋂
n∈N
(
1− 1
n
, 1 +
1
n
)
is
(a) {1} (b) (0, 2) (c) empty (d) None of these.
(4)
⋂
n∈N
(−n, n) is
(a) [−1, 1] (b) (−1, 1) (c) empty (d) None of these.
(5)
⋂
n∈N
[
− 1
n
,
1
n
]
is
(a) {0} (b) [−1, 1] (c) [0, 1] (d) None of these.
(6)
⋂
n∈N
[
0,
1
n
]
(a) {0} (b) empty (c) [0, 1] (d) None of these.
(7) f : R −→ R be any function (distance is usual). Then
(a) f is continuous on R if and only if f satisfies intermediate value property.
(b) If f is continuous on R then satisfies intermediate value property.
(c) If f satisfies intermediate value proerty and f−1({r}) is closed ∀ r ∈ Q then f is
continuous on R.
(d) None of the above.
(8) f : [0, 1] −→ [0, 1] is defined by
f(x) =
{
x if x ∈ Q ∩ [0, 1]
1− x if x ∈ (R \Q) ∩ [0, 1]
(a) f is continuous on [0, 1] and does not satisfy intermediate value property.
(b) f satisfies intermediate value property but f is not continuous.
(c) f is continuous only at x = 1
2
and f [0, 1] = [0, 1] . (d) None of the above.
(9) Cantor’s Theorem is applicable in the following and ∩n∈NFn is a singleton set
19
US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
(i) X = [−1, 1], d usual distance, Fn = [− 1n , 1n ]
(ii) X = (0, 1), d usual distance, Fn = [0,
1
n
]
(iii) X = R, d discrete metric, Fn = (0, 1n)
(iv) X = [0, 1], d usual distance, Fn = [1− 1n , 1]
(a) (i) and (ii) (b) (i) and (iv) (c) (i), (ii) and (iv) (d) None of these.
(10) Let d1 and d2 be equivalent metrics on X. Then
(a) (X, d1) is bounded =⇒ (X, d2) is bounded.
(b) (X, d1) is complete =⇒ (X, d2) is complete.
(c) (xn) is a Cauchy sequence in (X, d1) =⇒ (xn) is a Cauchy sequence in (X, d2).
(d) None of the above.
(11) Consider the following subspaces of R where distance in R is usual.
(i) Q (ii) Z (iii) {0}∪ { 1
n
: n ∈ N} (iv) [−1, 1)∪N. Then
(a) (i) and (iv) are complete .
(b) only (ii) is complete.
(c) (ii), (iii) and (iv) are complete.
(d) None of the above.
(12) Suppose ‖ ‖1 and ‖ ‖2 are equivalent norms on a normed linear space X. Then the
statement which is not true is
(a) (X, ‖ ‖1) is complete if and only if (X, ‖ ‖2) is complete.
(b) (xn) is a Cauchy sequence in (X, ‖ ‖1 if and only if (xn) is a Cauchy sequence in
(X, ‖ ‖2).
(c) A is a bounded set in (X, ‖ ‖1) if and only if A is bounded in (X, ‖ ‖2).
(d) (a), (b) and (c) are not true.
(13) Consider the following subspaces of (R, d) where d is usual distance :
(i) [0,∞) (ii) [0, 1] ∪ [2, 3] ( iii){1
1
, 1
2
, 1
3
, 2
3
, 1
4
, 3
4
, 1
5
, 2
5
, 3
5
, 4
5
, . . .} (iv) Z Then
(a) All the sub spaces are complete . (b) Only (i) is complete.
(c) Only (ii) is complete (d) Only (iii) is not complete.
(14) Let (X, d) be a complete metric space. A,B be complete subspaces of X such that
A ∩B 6= ∅ then
(a) A ∪B is a complete subspace of X but A ∩B is not.
(b) A ∩B is a complete subspace of X but A ∪B is not.
(c) A ∪B and A ∩B are complete subspaces of X.
(d) None of the above.
(15) Consider the following subspaces under usual distance in R.
(i) {√2,√3,√5} (ii) {√p : p is a prime number} (iii) {x ∈ R \Q : x ≤ √89} Then
(a) (i), (ii), (iii) are not complete.
(b) (i), (ii), (iii) are all complete.
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US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
(c) (i) and (ii) are complete and (iii) is not.
(d) None of the above.
(16) Consider the following subspaces of (R, d), where d is usual distance in R. If N,Z,Q,R\Q
are subspaces of (R, d). Then
(a) N,Z and Q are complete , R \Q is not complete.
(b) N,Z,Q and R \Q are all complete.
(c) N,Z are complete and Q,R \Q are not complete.
(d) None of the above.
(17) Consider the space C [a, b] with norms ‖ ‖1 and ‖ ‖∞ where ‖f‖1 =
∫ b
a
|f(x)| dx and
‖f‖∞ = sup{|f(x)|x ∈ [a, b]}. Then
(a) (C [a, b], ‖ ‖1 ) and (C [a, b], ‖ ‖∞) are complete.
(b) (C [a, b], ‖ ‖1 ) is complete but (C [a, b], ‖ ‖∞) is not complete.
(c) (C [a, b], ‖ ‖∞) is complete but (C [a, b], ‖ ‖1 ) is not complete.
(d) None of the above.
Topology of Metric Spaces: Practical 3.5
Complete Metric Spaces
DESCRIPTIVE QUESTIONS 3.5
(1) Check whether Cantor’s Intersection theorem is applicable for the following examples. Also,
find ∩n∈NFn in each case, where (Fn) is a sequence of subsets of R and the distance in R is
usual.
(a) Fn = (0,∞) (b) Fn = (0, 1n) (c) Fn = [1− 1n , 2 + 1n ]
(2) Let f : R −→ R be a function which satisfies intermediate value property: for a, b ∈ R
with f(a) < λ < f(b), there exists c between a and b such that f(c) = λ. Further if
{x ∈ R : f(x) = r} is closed set for each r ∈ Q, then show that f is continuous on R.
(3) Prove that there is no continuous function f : [0, 1] −→ R satisfying x ∈ Q⇐⇒ f(x) /∈ Q.
(4) Let f : R −→ R be a function such that f−1({x}) has exactly two points for each x ∈ R.
Show that f cannot be continuous on R.
(5) Let h be defined on [0, 1] (usual distance) as follows:
h(x) =

0 if x is irrational.
1
n
if x is rational numberm
n
,with(m,n) = 1
1 if x = 0
Prove that h is continuous only at irrational points in [0, 1].
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US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
(6) f : [0, 1] −→ [0, 1] is defined by
f(x) =
{
x if x ∈ Q ∩ [0, 1]
1− x if x ∈ R \Q ∩ [0, 1]
Show that f([0, 1]) = [0, 1] whereas f does not satisfy intermediate value property.
(7) Show that the equation cos x = x has at least one solution.
(8) Show that the equation x3 − 15x+ 1 = 0 has 3 solutions in the interval [−4, 4].
(9) Show that the function f(x) = (x− a)2(x− b)2 +x takes the value (a+ b)/2 for some value
of x.
(10) Let f(x) = tan x ; then f(pi/4) = 1 and f(3pi/4) = −1. But there is no c ∈ (pi/4, 3pi/4)
such that f(c) = 0. Explain why this does not contradicts Intermediate value property.
(11) Prove that if f, g are continuous on [a, b] and f(a) > g(a) and f(b) < g(b) then there is a
point c ∈ (a, b) such that f(c) = g(c).
(12) Use the intermediate value property to show that there is a square whose diagonal has
length between r and 2r and has area equal to half the area of the circle of radius r.
(13) Show that a Cauchy sequence in a metric space (X, d) where, X is a finite set and d is
any distance, is eventually constant. Hence show that (X, d) is complete.
(14) Show that Cauchy sequence in (N, d) (or (Z, d)) where d is usual distance is eventually
constant. Hence show that (N, d) (or (Z, d)) is complete.
(15) Show that a Cauchy sequence in a discrete metric space (X, d) is eventually constant.
Deduce that (X, d) is complete.
(16) Show that (R2, d) is a complete metric space where d(x, y) = 2|x1 − y1| + 3|x2 − y2| for
x = (x1, x2), y = (y1, y2) ∈ R2.
(17) Show that (N, d) is a complete metric space where for m,n ∈ N,
d(m,n) =
{
0 if m = n
1 +
1
m+ n
if m 6= n
(18) Let (X1, d1) and (X2, d2) be metric spaces and d be a metric on X1 × X2 defined by
d
(
(x1, x2), (y1, y2)
)
=
√
d21(x1, y1) + d
2
2(x2, y2). Show that
(
xn
)
=
(
x1(n), x2(n)
)
in X1 ×
X2 converges to (p1, p2) if and only if x1(n) −→ p1 and x2(n) −→ p2. Hence prove that if
X1, X2 are complete, then X1 ×X2 is complete.
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US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
(19) Let (X1, d1) and (X2, d2) be complete metric spaces. Show that
(
X1×X2, d′
)
and
(
X1×
X2, d
′′
)
are complete metric spaces where
d′
(
(x1, x2), (y1, y2)
)
= αd1(x1, y1) + βd2(x2, y2)
d′′
(
(x1, x2), (y1, y2)
)
=
√
αd21(x1, y1) + αd
2
2(x2, y2). where α, β > 0.
(20) Show that the metric space (C[0, 1], d1) is not complete where d1(f, g) =
∫ 1
0
|f(x) −
g(x)| dx.
Hint: Consider the sequence {fn} in C[0, 1] defined by
fn(t) =

0 if 0 ≤ t ≤ 1
2
− 1
n
nt− n
2
+ 1 if 1
2
− 1
n
< t ≤ 1
2
1 if 1
2
< t ≤ 1
(21) Prove that (0, 1) as a subspace of (R, d) (d being usual distance) is not complete but is
complete as a subspace of (R, d1) where d1 is discrete metric.
(22) Show that C[0, 1] with ‖ ‖∞ defined as ‖f‖∞ = sup{|f(t)| : t ∈ [0, 1]} is complete.
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US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
Topology of Metric Spaces: Practical 3.6
Compact Metric Spaces
Objective Questions 3.6
. (Revised Syllabus 2018-19)
(1) Let (X, d) be a metric space and K ⊆ X. Then
(a) K is compact. (b) K is compact if K is closed.
(c) K is compact if K is bounded. (d) K is compact if K is finite.
(2) Let (X, d) be a metric space and (xn) be a sequence in X such that xn → x0 as n → ∞.
Then
(a) {xn : n ∈ N} is a compact subset of X
(b) {xn : n ∈ N} ∪ {x0} is a compact subset of X
(c) {xn : n ∈ N} ∪ {x0} is a compact subset of X only if (xn) is a sequence of distinct
points.
(d) None of the above.
(3) Let {An} be a family of compact subset of a metric space (X, d) such that ∩n∈NAn 6= ∅.
Then
(a) A1 ∪ . . . ∪ Ak, k ∈ N and ∩n∈NAn are compact subsets of X.
(b) A1 ∩ . . . ∪ Ak, k ∈ N and ∪n∈NAn are compact subsets of X.
(c) ∪n∈NAn and ∩n∈NAn are compact subsets of X
(d) None of the above.
(4) Which of the following statements is false?
(a) A compact subset of a metric space is closed and bounded.
(b) A closed and bounded subset of a metric space is compact.
(c) A finite subset of a metric space is compact.
(d) A closed subset of a compact set in a metric space is compact.
(5) Which of the following are compact sunsets in the given metric space?
(a) [0, 1] in (R, d1) where d1 is discrete metric.
(b) N in (R, d) where d is usual distance.
(c)
{(
1
n
,
(−1)n
n
)
: n ∈ N
}
∪ {(0, 0)} in (R2, d) where d is Euclidean distance.
(d) [a, b] ∩Q where a, b are irrational numbers in (Q, d) where d is usual distance.
(6) Consider the following subsets of (R2, d), (d being Euclidean distance)
(i) A = {(x, y) ∈ R2 : x2 − y2 = 1}
(ii) B = {(x, y) ∈ R2 : y2 = x}
(iii) C = {(x, y) ∈ R2 : 2x2 + 3y2 = 100} Then
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US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
(a) A,B,C are compact.
(b) B,C are compact and A is not compact.
(c) Only A,B are compact.
(d) C is compact.
(7) Let (X, d) be a metric space and x ∈ X. Let B[x, r] denote the closed ball {y ∈ Y :
d(x, y) ≤ r} Then
(a) B[x, r] is compact. (b) B[x, r] is compact only if r ≤ 1.
(c) B[x, r] is compact if X = R and d is Euclidean distance. (d) None of the above.
(8) In the metric space (Z, d), (Z is the set of integers, d is usual distance), K ⊂ Z
(a) if and only if K is closed. (b) if and only if K is bounded.
(c) if and only if K has a limit point. (d) if and only if 0 ∈ K.
(9) Which of the following subsets of R3 are compact?
(a) {(x, y, x) ∈ R3 : x2 + y2 − z2 = 1} (b) {(x, y, x) ∈ R3 : x2 − y2 − z2 = 1}
(c) {(x, y, x) ∈ R3 : x2 + y2 + z2 = 1} (d) None of the above.
(10) Which of the following subsets of R2 is not compact? (distance being Euclidean) (a) The
ellipse {(x, y) ∈ R2 : x
2
a2
+
y2
b2
= 1}, (a, b > 0)
(b) The rectangular hyperbola {(x, y) ∈ R2 : xy = 1}
(c) The set {(x, y) ∈ R2 : x2 + 2y2 ≤ 32} (d) The set {(x, y) ∈ R2 : |x| ≤ 1, |y| ≤ 1}
(11) In the metric space (R, d) (d begin usual distance)
(a) [0, 1] ∪ [2, 3] is compact. (b) [0, 1] ∪ (2, 3) is compact.
(c) [0, 1] ∪ {x ∈ N : x ≥ 3} is compact. (d) [0, 1] ∪ [2,∞) is compact.
(12) Consider the following subsets of R2 (distance being Euclidean).
(i) A = {(x, y) ∈ R2 : x2 + y2 = 1}
(ii) B = {(x, y) ∈ R2 : x2 + y2 ≤ 1}
(iii) C = {(x, y) ∈ R2 : x2 + y2 ≥ 1}
(a) A,B,C are all compact. (b) A and B are compact, C is not compact.
(c) Only B is compact. (d) Only A is compact.
(13) Consider the following subsets of (Rn, d) (d being Euclidean distance)
A = {(x1, . . . , xn) ∈ Rn : x1 + x2 + . . .+ xn = 0}
B = {(x1, . . . , xn) ∈ Rn :
n∑
i=1
x2i = 1}
C = {(x1, . . . , xn) ∈ Rn :
n∑
i=1
|xi| ≤ n for 1 ≤ i ≤ n}
D = {(x1, . . . , xn) ∈ Rn : x1 = xn = 0}
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US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
(a) A,B,C,D are compact sets. (b) Only B and C are compact sets.
(c) Only B,C and D are compact sets. (d) None of the above.
(14) Let A,B be compact subsets of (R, d), (d being usual). Then the following set is not
compact.
(a) A×B in (R2, d), d being Euclidean (b) A ∪B in R
(c) A ∩B in R (provided A ∩B 6= ∅). (d) A \B in R (provided A \B 6= ∅).
(15) Let (xn) be a sequence in [0, 1]. Then, which of the following is not true.
(a) (xn) has a convergent subsequence.
(b) (xn) is bounded but may not be convergent.
(c) (xn) is Cauchy.
(d) (xn) may have subsequences converging to different limits.
(16) Let A be a compact subset of R. Then
(a) A may not be compact. (b) A◦ may not be compact.
(c) ∂A may not be compact. (d) None of the above.
(17) Let A be a compact subset of R. Then which of the following statements is not true
(a) A is complete. (b) A has a limit point in R
(c) A is closed and bounded. (d) A◦ and ∂A are bounded.
Topology of Metric Spaces: Practical 3.6
Compact Metric Spaces
Descriptive Questions 3.6
(1) Using definition, show that K =
{
1
n
: n ∈ N
}
∪ {0} is a compact subset of (R, d), where d
is usual distance in R. Also find a finite subcover of the open cover {B( 1
n
, 1
10
)}n∈N of K.
(2) Let (X, d) be a metric space and (xn) be a sequence in X converging to x0. Using definition,
show that K = {xn : n ∈ N} ∪ {x0} is a compact subset of (X, d)
(3) In the following examples, show that the set is not compact b considering the given open
cover of the set:
(i) C [a, b] in the metric space (C [a, b], ‖ ‖∞), ‖f ‖∞ = sup {|f (t)| : t ∈ [a, b]}. Show that
the open cover {B(0, n)}n∈N of C [a, b] has no finite subcover. 0 being the constant
zero function).
(ii) (0, 1) in the metric space (R, d), d being the usual distance . Show that the open cover
{( 1
n
, 1)}n∈N of (0, 1) has no finite subcover.
(iii) { 1
n
: n ∈ N} in the metric space (R, d), d being the usual distance. Show that the open
cover {( 1
2n
, 3
2n
)}n∈N of { 1n : n ∈ N} has no finite subcover.
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US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
(iv) [0, 1] in the metric space (R, d1), d1 being the discrete metric. Show that the open
cover {B(x, 1
2
)}x∈[0,1] has no finite subcover.
(4) Check if the following sets are compact in the given metric space. Justify your answer.
(i) {(x, y) ∈ R2 : x2 − y2 = 1} in (R2, d), d being Euclidean metric.
(ii) {(x, y) ∈ R2 : xy = 1} in (R2, d), d being euclidean metric.
(iii) {n+ 1
n
: n ∈ N} in (R, d), d being usual distance.
(5) Prove or disprove:
(i) A closed and bounded subset of a metric space is compact.
(ii) A closed ball B[x, r] in a metric space is compact.
(iii) A compact set in a metric space is not open.
(iv) Interior and closure of a compact set are compact.
(6) Determine which of the following subsets of (R2, d), where d is Euclidean distance is com-
pact. Justify your answer.
(i) {(x, y) ∈ R2 : |x|+ |y| ≤ 1}
(ii) {(x, y) ∈ R2 : |x| ≤ 1}
(iii) {(x, y) ∈ R2 : x ≥ 1, 0 ≤ y ≤ 1
x
}
(iv) {(x, y) ∈ R2 : x2
a2
+ y
2
b2
= 1}, (a, b > 0)
(v) {(x, y) ∈ R2 : xy = 0}
(7) Let A,B be compact subsets of R, distance being usual. Show that
(i) A+B is a compact subset of R.
(ii) A ∪B is a compact subset of R.
(iii) A×B is a compact subset of (R2, d), d being Euclidean distance.
(8) Show that A = (0, 1] is not a compact subset of (R, d), d being Euclidean distance by
(i) exhibiting a sequence in A which has no convergent sequence.
(ii) exhibiting an infinite subset of A which has no limit point in A.
(9) Show that {(x1, x2, . . . , xn) ∈ Rn : x21 + 2x22 + · · ·+ nx2n ≤ (n+ 1)2} is a compact subset of
(Rn, d), d being Euclidean.
(10) If A,B are disjoint non-empty subsets of (X, d) and A is closed, B is compact then show
that d(A,B) > 0.
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US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
(11) Consider the set A = (−√2,√2) ∩ Q in a metric space (Q, d) where d is a usual metric
from R. Is the set A:
(i) closed and bounded in (Q, d)?
(ii) compact in (Q, d)?
(12) Show that the closed unit ball B[0, 1] in l2 is not compact, where l2 := {(xn) in R :
∞∑
n=1
|xn|2 <∞ i.e. convergent }; Further, for any x = (xn) ∈ l2; define ||x||2 =
√√√√ ∞∑
n=1
|xn|2.
The metric on l2 is the metric corresponding to this norm.
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US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
Topology of Metric Spaces: Practical 3.7
Miscellaneous.
. Revised Syllabus 2018-19
UNIT 1
(1) Define a metric space (X, d) and a normed linear space (X, ‖ ‖). Show that on a normed
linear space d : X ×X −→ R defined by d(x, y) = ‖x− y‖ is a metric.
(2) Define an open ball B(x, r) in a metric space (X, d). Show that an open ball is an open
set.
(3) State and prove Hausdorff property in a metric space (X, d)
(4) Show that in a metric space (X, d)
(i) an arbitrary union of open sets is open.
(ii) a finite intersection of open sets is open.
(5) Give an example to show that an arbitrary intersection of open sets need not be open.
(6) Let (X, d) be a metric space. Show that a subset G of X is open if and only if it is a union
of open balls.
(7) Prove that any nonempty open subset of R (distance being usual) can be written as a finite
or countable union of open mutually disjoint intervals.
(8) Let (X, d) be a metric space and A ⊆ X. Show that
(i) A◦ is an open set and is the larges open set contained in A.
(ii) A is open if and only if A = A◦
(9) Let (X, d) be a metric space and A ⊆ X. Show that
(i) A ⊆ B =⇒ A◦ ⊆ B◦
(ii) A ∩B)◦ = A◦ ∩B◦
(iii) A◦ ∪B◦ ⊆ (A ∪B)◦ and the inequality may be strict.
(10) Show that two metrics d and d′ on a non-empty set X are equivalent if and only if for
each x ∈ X, any open ball Bd(x, r) contains an open ball Bd(x, r,′ ) for some r′ > 0 and
any open ball B′d(x, s) contains an open ball Bd(x, s
′) for some s′ > 0.
(11) Let (X, d) be a metric space and F be a subset of X. Show that the following statements
are equivalent:
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US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
(i) X \ F is open.
(ii) F contains all its limit points.
(12) Show that in a metric space (X, d) , the following statements are equivalent for a subset
G of X.
(i) G is open
(ii) G does not contain any limit point of X \G.
(13) Let (X, d) be a metric space and A ⊆ X. Show that
(i) A is a closed set.
(ii) A is closed if and only if A = A.
(14) Let (X, d) be a metric space and A,B ⊆ X. Show that
(i) A ⊆ B =⇒ A ⊂ B
(ii) A ∪B = A ∪B
(iii) A ∩B ⊆ A ∩B and the inequality may be strict.
(15) Let (X, d) be a metric space and A ⊆ X. Show that D(D(A)) ⊆ D(A) where D(S) denotes
the set of limit points of S ⊆ X. Hence show that D(A) is closed.
(16) Bolzano-Weierstrass Theorem: Consider a metric space (R, d), where d is the usual metric.
Prove that every infinite bounded subset of R must have a limit point in R.
UNIT II
(1) Let (X, d) be a metric space and A ⊆ X. Show that p ∈ A if and only if there is a sequence
of points in A converging to p.
(2) Let (X, d) be a metric space and A be a subset of X. Show that p is a limit point of A if
and only if there is a sequence of distinct points converging to p.
(3) Prove: Every bounded sequence in R with usual metric, has a convergent subsequence.
(4) Show that a sequence (xk) in (Rn, d) (where d is Euclidean distance) converge to a point
p = (p1, p2, . . . , pn) ∈ Rn if and only if xik −→ pi, for 1 ≤ ni in R with respect to the usual
distance, where xk = (x
1
k, x
2
k, · · · , xnk). Hence deduce that (Rn, d) is a separable metric
space.
(5) Let (X, d) be a metric space and Y be a non-empty subset of X. Show that
(i) A subset G of Y is open in the subspace (Y, d) if and only of G = V ∩ Y where V is
an open set in (X, d)
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US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
(ii) A subset F of Y is closed in the subspace (Y, d) if and only if F = H ∩ Y where H is
closed set in (X, d).
(6) Let (X, d) be a metric space. Show that a convergent sequence in (X, d) is Cauchy. Give
an example to show that the converse is not true. further show that a Cauchy sequence
(xn) in (X, d) is convergent if and only if it has a convergent subsequence.
(7) Show that the metric spaces (X, d1) and (X, d2) are equivalent if and only if (xn) converges
to p in (X, d1) if and only if (xn) converges to p in (X, d2)
(8) Let (X, d) be a metric space . Show that a subset A of X is dense in X if and only if
G ∩ A 6= ∅ for each non-empty open subset G of X.
(9) Let (X, d) be a metric space. If A ⊆ X is dense in X and B is a non-empty open subset of
X then A ∩B = B.
(10) Prove that the metric space (R, d) is complete where d is the usual distance.
(11) Prove that the metric space (R2, d) is complete where d is the Euclidean distance.
(12) Prove that the metric space (C, d) is complete with respect to the distance given by
d(z1, z2) = |z1 − z2|
(13) Show that the metric space (C[a, b], d) is complete where d(f, g) = sup{|f(x)− g(x)| : x ∈
[a, b]}.
(14) Let (X, d) be a metric space and (Y, dY ) be a subspace of (X, d). If (Y, dY ) is complete
then show that Y is closed.
(15) Let (X, d) be a complete metric space. If Y is a closed subspace of X then show that the
subspace (Y, dY ) is complete.
(16) State and prove Cantor’s intersection theorem in a metric space (X, d).
(17) If in a metric space (X, d), for every decreasing sequence {Fn} of non-empty closed sets
with d(Fn) −→ 0,∩n∈NFn is a singleton set then prove that (X, d) is complete.
(18) Nested Interval Theorem (As a particular case of Cantor’s intersection theorem): Let
Jn = [an, bn] be a sequence of intervals in R such that Jn+1 ⊆ Jn∀n ∈ N. Then show
that
⋂
n∈N
Jn 6= ∅. If further we assume that lim
n−→∞
`(Jn) = 0 then show that
⋂
n∈N
Jn contains
precisely one point.
As a consequence of Nested Interval Theorem:
(19) Show that set R of real numbers is uncountable.
(20) Density of rationals: Let x < y be real numbers. Show that there exists a rational number
r ∈ Q such that x < r < y.
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US/AMT503 Sem V, Paper 3: Topology of Metric Spaces Revised Syllabus 2018
(21) Intermediate Value Theorem: Let f : [a, b] −→ R be continuous. Assume that f(a) and
f(b) are of different signs, say, f(a) < 0 and f(b) > 0. Show that there exists c ∈ (a, b)
such that f(c) = 0.
UNIT III
(1) Show that a compact subset of a metric space is closed and bounded. Give an example to
show that a closed and bounded subset of a metric space is not compact.
(2) Prove: A closed subset of a compact metric space is compact.
(3) Let (X, d) be a metric space and K is a compact subset of X. If F is a closed subset of X
then show that F ∩K is compact.
(4) Suppose (X, d) is a metric space and C is a non-empty collection of compact subsets of X
then
(i)
⋂
K∈C
K is a compact subset of X.
(ii) If C is finite then ⋃
K∈C
K is a compact subset of X.
(5) Prove that a set A in a discrete metric space (X, d) is compact if and only if A is a finite
set.
(6) Consider a metric space (R, d) where d is usual metric, ∅ 6= A ⊂ R. Prove that A is closed
and bounded if and only if A satisfy Hein-Borel property. (A set is said to satisfy Hein-
Borel property if every open conver of that set admits finite subcover).
Remark: The above result can be generalised to (Rn, d) as follows(without proof):
A subset A of (Rn, d) is closed and bounded if and only if it satisfy Hein-Borel property.
Hence, A ⊂ Rn is compact if and only if it is closed and bounded.
(7) Consider a metric space (R, d) where d is usual metric, ∅ 6= A ⊂ R. Prove that A is closed
and bounded if and only if A is sequentially compact. (A set A is said to be sequentially
compact if every sequence in A has a covergent subsequence).
(8) Consider a metric space (R, d) where d is usual metric, ∅ 6= A ⊂ R. Prove that A is
sequentially compact if and only if A satisfy Bolzano-Weierstrass property. (A set A is said
to satisfy Bolzano-Weierstrass property if every non-empty, infinite subset of A has a limit
point in A).
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