MATH 131: MIDTERM
Due Friday October 21. The midterm is to be completed individually. You may use
Munkres Topology, but other books or electronic references should not be consulted.
1. Let X be a metrizable topological space. Show that X is compact if and
MATH 131: EXAMPLE CORRECTED FROM CLASS
Let RZ denote the set of functions from Z to R with the uniform topology, i.e. with
the topology induced from the metric (f, g) = supnZ (min(1, |f(n) g(n)|). Recall the
notation B(f, ) = cfw_g|(f, g) < .
Let fk : Z R
Two brainteasers
Here are two topology brainteasers to think about over the weekend. The rst is about 100 years old,
and I talked about it in section earlier this week.
Problem 1.
1. You have a topological space X and a bag of subsets of X, which starts o
MATH 131: FINAL SOLUTIONS
1. Let X be a topological space and let A be a subspace. Let i : A X denote the inclusion
map. A is said to be a deformation retract of X if there exists a continuous map f : X A such
that f(a) = a for all a A and a homotopy betw
MATH 131 SOLUTION SET, WEEK 2
1. Characterisation of open sets in R
Here are two solutions. The rst is the one Alex and I knew before speaking to Prof.
McMullen a couple of days ago, and is the one we hinted at if you came to oce hours or
section. The sec
MATH 131 SOLUTION SET, WEEK 1
1. not from Munkres
Problem 1.1. One sentence solution: Qn is countable and dense in Rn .
So lets rst show this:
Lemma 1.1. |N| = |Qn | for any n Q.
Proof. Typical solutions cited a theorem in Munkres that says that a nite pr
MATH 131 SOLUTION SET, WEEK 3
1. not Munkres
1. Let A X, and Recall the following characterization of the closure:
Fact 1.1 (Theorem 17.5 from Munkres). For a space X and A X, x A i every neighborhood
of x intersects A.
Thus,
x A x int(A) int(X A)
x int(
MATH 131: FINAL
Due Sunday December 11 by 5pm. Please bring your nal to room 239, or send it to me
over e-mail. If I am not in, slip your exam under my door.
The nal is to be completed individually. You may use Munkres Topology, but other
books or electro
MATH 131: MIDTERM SOLUTIONS
1. Proposition. Let X be a metrizable topological space. Show that X is compact every
continuous function f : X R is bounded.
Proof. : let f : X R be a continuous function. Since X is compact, f(X) is compact
(Munkres Ch. 3, 26
Exercises on Stones Representation Theorem
1
Introduction
Stones representation theorem is one of the things people point to when theyve had a few too many drinks
and are talking about the duality between topology and algebra. Im not convinced that such a
MATH 131 SOLUTION SET, WEEK 4
0.1. Notation. In problems 1 and 2, I will denote the origin in Rn by 0.
1. Topologists sine curve
Let f : (0, 1] R2 given by x (x, sin(1/x), and g : [1, 1] R2 given by
y (0, y). Then the topologists sine curve is by denition