# MATH4530–Topology. HW5 solutions

#### Document technical information

Format pdf
Size 62.3 kB
First found May 22, 2018

#### Document content analysis

Category Also themed
Language
English
Type
not defined
Concepts
no text concepts found

#### Transcript

```MATH 4530 – Topology. HW 5 solutions
Please declare any collaborations with classmates; if you find solutions in books or online,
Write the proofs in complete sentences.
(1) Show that any infinite set with the finite complement topology is connected.
Solution: If X = C1 t C2 where C1 , C2 are non-empty closed sets, since C1 and C2 must be
finite, so X is finite. This is a contradiction.
(2) Let T and T 0 be topologies on a set X. If T is finer than T 0 , then the connectedness of which
Solution: We have a condituous map idX : (X, T ) → (X, T 0 ). Since the image of a connected space is connected, the connectedness of T implies T 0 .
(3) Let p : X → Y be a quotient map. Show that, if p−1 (y) is connected for each y ∈ Y and Y is
connected, then X is connected.
Solution: Suppose that X is not connected. Say, X = U1 t U2 for some non-empty open
sets U1 and U2 . By Theorem 5.5, every p−1 (y) is contained in either U1 or U2 . Let V1 =
{y ∈ Y | p−1 (y) ⊂ U1 } and V2 = {y ∈ Y | p−1 (y) ⊂ U2 }. Since U1 and U2 are non-empty,
disjoint and p is surjective, we have Y = V1 t V2 and V1 and V2 are non-empty. Also,
π−1 (V1 ) = U1 , p−1 (V2 ) = U2 by surjectivity of p, so by the definition of quotient maps, V1
and V2 are open sets in Y. It follows that Y is not connected. Contradiction.
(4) Let f : X → Y be a continuous map. Show that if X is path-connected, then Im f is pathconnected.
Solution: Let x, y ∈ Im f . Let x1 ∈ f −1 (x) and y1 ∈ f −1 (y). Since X is path connected,
there is a path p : [0, 1] → X connecting x1 and y1 . Then f ◦ p is a path connecting x and y.
(5) Show that there is no homeomorphism between (0, 1) and (0, 1] by using the connectedness. Hint:
if we remove a point from each of the spaces, what happens?
Solution: Suppose that there is a homeomorphism f : (0, 1] → (0, 1). If f : X → Y is a
homeomorphism, then the restriction f |X−{x} : X − {x} → Y − { f (x)} is a homeomorphism
too. If we apply this to (0, 1] − {1} = (0, 1), then we have a homeomorphism (0, 1) (0, 1) − { f (1)}. LHS is connected but the right hand side is (0, f (1)) t ( f (1), 1) which is
disconnected. So we have a contradiction.
(6) Show that if X is connected, then any continuous map f : X → Y where Y is a topological space
with discrete topology is a constant map, i.e. f (X) = {y} for some y ∈ Y.
Solution: If f (X) has more than one element, then since Y has the discrete topology, f (X)
is disconnected. Since the image of a connected space must be connected, we have a contradiction. Thus f (X) has only one element, i.e. f is a constant map.
1
2
(7) Consider the quotient space of R2 by the identification (x, y) ∼ (x + n, y + n) for all (n, m) ∈ Z2 .
Show that it is connected and compact.
Solution: Since R2 is conencted, the quotient space must be connencted since the quotient
space is the image of a quotient map from R2 . Consider E := [0, 1] × [0, 1] ⊂ R2 , then the
restriction of the quotient map p : R2 → R2 / ∼ to E is surjective. Since [0, 1] × [0, 1] is
compact, the image R2 / ∼ must be compact.
(8) Recall that a square matrix M is orthogonal if MM t = I. This condition is equivalent to “the set
~ i = h~v, w
~ i. In particular, if M is
of row vectors form an orthonormal basis” and also to hM~v, M w
orthogonal, then det M = ±1. Let O(n, R) be the set of orthogonal matrices of size n. Show that it
is not connected.
Solution: The determinant of an orthogonal matrix is ±1. Since M1 := {M : det M = 1}
and M−1 := {M : det M = −1} are closed subsets of the set of matrice, we have
O(n, R) = (M1 ∩ O(n, R)) t (M−1 ∩ O(n, R))
is a disjoint union of closed sets. Each of them is non-empty, since we have the identity
matrix In ∈ M1 ∩ O(n, R) and the identity matrix with −1 multiplied to exactly one of the
diagonal entry is in M−1 ∩ O(n, R).
References
[M]
[S]
[L]
Munkres, Topology.
Basic Set Theory, http://www.math.cornell.edu/∼matsumura/math4530/basic set theory.pdf
Lecture notes, available at http://www.math.cornell.edu/∼matsumura/math4530/math4530web.html
```