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Math 101 — Final Exam
December 4–5, 2017
Name:
- This exam has 7 pages including this cover. There are 5 problems. Note that the problems are not of equal difficulty, so you may want to skip over and return to a problem on which you are stuck.
- In your solutions, you may refer to any of the theorems proved in class or on the homework.
- You may not use any external sources on this exam other than the course ma- terials (class notes and problem sets). You may not use any textbooks or external websites and you may not discuss the problems with anybody.
- This exam is due at 6 PM on December 5 in my office (SC 503).
- Make sure you read and sign the last page before turning in the exam.
Problem Points Score
Total 40
- [6 points]
a. [4 points] Let X be a finite set. Let T be a topology on X. Define
T ′^ = {S | X − S ∈ T }.
Show that T ′^ is a topology on X.
b. [2 points] Give a counterexample to show that the statement in part a. is false if X can be infinite.
- [10 points] Give an example of each of the following. You do not need to prove that your example works. a. [2 points] A subset of R that is neither open nor closed in the lower limit topology.
b. [2 points] A group G such that |G| > 2 and if H ≤ G, then H = {e} or G. i.e. G has no proper nontrivial subgroups.
c. [2 points] A set S along with an order relation such that every element has an immediate successor but at least one element (other than the smallest element) doesn’t have an immediate predecessor.
d. [2 points] A topology T on the set X = {a, b, c, d} such that {a, b}, {b, c} ∈ T , but T is not the discrete topology.
e. [2 points] A basis for a topology on R that is not comparable to the standard topology.
- [8 points] Let T be the standard topology on R. Define
T ′^ = T ∪ {U ∪ { 0 } | U ∈ T }.
a. [4 points] Check that T ′^ is a topology on R.
b. [4 points] Consider R equipped with the topology T ′. Let A ⊆ R. Show that 0 ∈ A¯ if and only if 0 ∈ A.
Before handing in your exam, please read the following and sign below.
While completing this exam, I have not consulted any external sources other than class notes and problem sets. I have not discussed the problems or solutions of this exam with anyone.
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