Math 434/542
Fall 2015
Final Exam, Dec. 7, 2015
Carter
Instructions: Attempt all of Section 1. Work at least 3 problems from Section 2. If in
Section 2, you don’t recall the definitions correctly, then turn in Section 1, and ask me for a
definition sheet. Do at least 3 problems from Section 3. Attempt both problems in Section 4.
Thank you for your participation in this class. It has been a pleasure to watch you learn.
1
Basic Definitions.
Give your most correct definition of the terms that follow.
1. Define a topological space.
2. Define a Hausdorff space.
3. Define a compact space.
4. Define a metric space.
5. Define a continuous function.
6. Define a homeomorphism.
2
Basic Proofs.
1. Prove that a metric space is Hausdorff.
2. Prove that a compact subset of Hausdorff space is closed.
3. Prove that the continuous image of a compact space is compact.
f
4. Prove that if X is a compact Hausdorff space, then a real-valued function R ←− X
achieves its optimal values.
3
Some More Advanced Problems
1. Consider the “eye-glass” graph and the “theta-curve” graph that are illustrated. Prove
or disprove that they are homeomorphic.
Eye-Glass Graph
Theta-curve graph
1
2. Use the arc-tangent function to prove that the real line R and the open interval
(−1, 1) = {x ∈ R| − 1 < x < 1}. You will also have to rescale. Give as many
details as possible.
3. Give a picture proof that the square [−1, 1] × [−1, 1] = {(x, y) ∈ R2 | − 1 ≤ x, y1} and
the closed disk D2 = {(x, y) ∈ R2 |x2 + y 2 ≤ 1} are homeomorphic.
4. Prove that the real line R and the plane R × R = R2 are NOT homeomorphic.
f
5. Prove that any function [0, 1] ←− [0, 1] has a fixed point f (x) = x.
4
Classification Theorem of Surfaces
1. Consider the decagon with edges identified in pairs as indicated. Compute its Euler
characteristic, χ(F ). Use the formula g = 2−χ(F ) to compute the genus: the number
of toral connect summands — the number of holes in the surface. If you have time,
apply the proof to put the surface into normal form.
a
b
d
e
a
d
c
c
e
b
2. Prove that a shirt and a pair of pants are not homeomorphic. Hint: How many arcs
are needed to separate each?
Help the Oompa-loompa get dressed. Are the shirt and pants
homeomorphic?
2