Homework 7
Math 351, Spring 2015
Due Date: Saturday, March 28
Instructions: In each of the following problems, find an explicit formula for the requested
function.
1. Let D2 be the disk {(x, y) ∈ R2 | x2 + y 2 ≤ 1}, and let X = D2 − {(0, 1), (0, −1)}.
Find a homeomorphism from X to the square [−1, 1] × (−1, 1).
2. Let X = {(x, y, z) ∈ R3 | z ≥ x2 + y 2 }. Find a homeomorphism from X to R2 × [0, 1).
3. Let T be the triangle {(x, y) ∈ R2 | x ≤ 1 and −x ≤ y ≤ x}, let P be the parallelogram
{(x, y) ∈ R2 | 0 ≤ x ≤ 1 and x − 1 ≤ y ≤ x}, and let S be the square [0, 1] × [0, 1].
(a) Find a homeomorphism f : T → P .
(b) Find a homeomorphism g : P → S.
(c) Find a homeomorphism h : T → S.
4. Find a continuous function f : S 1 → S 1 such that every point has exactly two preimages.