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 metr
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)|). Rec
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 topolo
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 :
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
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 soluti
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
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. Y
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.
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 bet
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