Takehome Final

Takehome Final
Math 351, Spring 2015
This final exam has four problems. Your solutions must be written up in LATEX, and are due
on Tuesday, May 19 at midnight. Late solutions will not be accepted.
Here are the rules for the midterm:
• You may use the textbook as a reference, but you should not consult any other books
on topology.
• You should feel free to use the materials on the class web page, but you may not consult
any other internet sources.
• You may not discuss the problems or their solutions with anyone but me.
If you are having trouble, feel free to e-mail me a question, or come to my office hours
sometime this week.
1. Prove that every compact metric space has a countable basis for its topology.
2. Let X be a metric space. Suppose that, for every pair of points p, q ∈ X, there exists
a path f : [0, 1] → X from p to q such that
d f (s), f (t) ≤ |s − t|
for all s, t ∈ [0, 1]. Prove that X ω is path connected in the uniform topology.
3. Prove that for every closed subset B of R2 , there exists a subset S of R2 whose boundary
is equal to B.
4. Let X be a compact space, and let Y be a connected Hausdorff space. Let p : X → Y
be a continuous surjection, and suppose that p−1 ({y}) is connected for each y ∈ Y .
Prove that X is connected.