More examples of Quotient Spaces
Topology MTH 441 Fall 2009
Abhijit Champanerkar
1. Let P = {{(x, y)| x − y = c}| c ∈ R} be a partition of R2 . Note that
P is a union of parallel lines. Describe the quotient space R2 / ∼.
2. Let X be a topological space and A ⊂ X. Let P be a partition of X
which consists of the sets A and {x} for x ∈ X − A. Let X/A denote
the quotient space with respect to this partition. In X/A, the set A is
identified to a point. Describe X/A and draw pictures for the following:
(a) X = {(x, y, z)| x2 + y 2 = 1} and A = {(x, y, 0)| x2 + y 2 = 1}.
(b) X = S 2 and A is the equator in the xy-plane.
3. (Torus) Let T 2 be the torus defined as a quotient space of the square.
Let b > a > 0. Consider the map F : [0, 1] × [0, 1] → R3 defined by
F (s, t) = ((b+a cos(2πt)) cos(2πs), (b+a cos(2πt)) sin(2πs), a sin(2πt)).
Show that:
(a) F is a quotient map onto its image.
(b) F factors to a map f from T 2 to R3 .
(c) f is a homeomorphism onto its image.
Note: The image of F is an imbedding of the torus in R3 thought of
as a boundary of a doughnut. The equations in F are obtained by a
parametrization of this torus as a surface of revolution of a circle of
radius a centered at (b, 0, 0) in the xz−plane and revolving about the
z−axis in counterclockwise direction. This shows that the torus as a
quotient space of [0, 1] × [0, 1] is homeomorphic to the torus we draw in
R3 . Since we know that T 2 is homeomorphic to S 1 × S 1 , this exercise
shows that the boundary of a doughnut is homemorphic to S 1 × S 1 .
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t
s
p
4. For x = (x1 , . . . , xn ) ∈ Rn let |x| = x21 + . . . + x2n . |x| is the length
of x. Let B n = {x ∈ Rn | |x| < 1 }. B n is the open unit ball in Rn in
the Euclidean metric. Define h : B n → Rn and k : Rn → B n by
h(x) =
x
,
1 − |x|
k(x) =
x
1 + |x|
Show that h and k are continuous and inverses of each other. This
shows that B n is homeomorphic to Rn .
5. (Spheres) Let S n = {x ∈ Rn+1 | |x| = 1 }. The n-sphere S n is the set
of all unit vectors in Rn+1 . Let Dn = {x ∈ Rn | |x| ≤ 1 }. Dn is the
closed unit ball in Rn . Note that the boundary of Dn is S n−1 . This
exercise will show that S n is obtained by identifying the boundary of
Dn to one point.
Let N = (0, . . . , 0, 1) ∈ S n be the north pole. Define f : S n − N → Rn
by
x1
xn
,...,
)
f (x) = (
1 − xn+1
1 − xn+1
This map is called the stereographic projection. Define g : Rn →
S n − N by
2x1
2xn
|x|2 − 1
g(x) = ( 2
,..., 2
, 2
)
|x| + 1
|x| + 1 |x| + 1
We can use any any point of S n instead of the north pole. The picture
below illustrates the stereographic projection for S 2 .
2
N
x
f(y)
f(x)
y
(a) Show that f and g are continuous and inverses of each other. This
shows that S n − N and Rn are homeomorphic.
(b) Define F : Dn → S n by
F (x) =
g(h(x)) if |x| < 1
N
if |x| = 1
Show that F is a quotient map.
(c) Show that Dn /S n−1 is homeomorphic to S n .
6. (Projective spaces) Define the following quotient spaces:
• For x ∈ S n , let x ∼ −x. Let X = S n / ∼.
• For x ∈ Dn , let x ∼ −x if |x| = 1 otherwise x ∼ x. Let Y =
Dn / ∼.
• For x, y ∈ Rn+1 − {0}, let x ∼ y if y = α x for some non-zero real
α. Let Z = Rn+1 / ∼.
Show that
(a) The spaces X, Y and Z are homeomorphic to each other. This
space is known as the n-dimensional real projective space and
denoted by RPn .
(b) RP2 is homeomorphic to the projective plane P 2 .
7. (Hawaiian earring) Let X = {(x, y)| (x − n1 )2 + y 2 = ( n1 )2 , n ∈ Z+ }.
X is called a Hawaiian earring and it consists of infinitely many circles
glued at one point. Let Y = R/Z. Y also consists of infinitely many
circles glued at one point. Draw pictures of both. Show that X is not
homeomorphic to Y . This is an example where the quotient space is
not homeomorphic to the picture imbedded in R2 .
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8. (Attaching maps) We can use continuous functions to glue two spaces to
each other. Let X and Y be topological spaces, A ⊂ Y and f : A → X
be a continuous function. Define a relation on X t Y (disjoint union
of X and Y ) as follows:
• a ∼ f (a) for a ∈ A
• x ∼ x for x ∈ X − f (A)
• y ∼ y for x ∈ Y − A
The quotient space (X t Y )/ ∼ is denoted by X ∪f Y and f is called
the attaching map. Find X ∪f Y for the following:
(a) X = Y = D2 , A = S 1 and f : S 1 → D2 is the inclusion.
(b) X be the Mobius band, Y = D2 , A = S 1 and f : S 1 → M be
the homeomorphism of S 1 with the boundary circle of the Mobius
band.
(c) For > 0 and very small, let B = {(x, y)| (x−1/2)2 +(y −1/2)2 <
} ⊂ [0, 1] × [0, 1]. Let X = Y = T 2 − B. Note that X and Y
are homeomorphic to a torus with a small open disc removed and
have a circle as boundary. Let A = ∂Y and let f : ∂Y → X be
the identification with the boundary of X.
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