1. Metrics

Define two distance functions d1, d2 : R^2 × R^2 → R on Euclidean plane R^2 by the formulas

• d1(x, y) = |x1 − y1| + |x2 − y2|
• d2(x, y) = max{|x1 − y1|, |x2 − y2|}

for x = (x1, x2), y = (y1, y2) ∈ R^2

• Check that both d1 and d2 are metrics on R2
• Make sketches of the unit disk about the origin with respect to these metrics.

2. Topology

Let X, Y be nonempty sets and f : X → Y be an arbitrary map. Given a topology T on Y , show that the family of sets
f^(−1)(T ) = {f^(−1)(V ) : V ∈ T }
is a topology on X.

3. Jordan curve

Recall the Jordan Curve Theorem which says that every simple closed curve in the plane decomposes R^2 into two.

• Show the same is true for a simple closed curve on the sphere, S^2 ={x ∈ R^2: ||x|| = 1}.
• Give an example that shows the result does not hold for simple closed curves on the torus.

4. Homeomorphisms

Give explicit homeomorphisms to show that the following spaces with topologies inherited from the respective containing Euclidean spaces are homeomorphic:

• R, the real line;
• (0, 1), the open interval;
• S^1 − {(0, 1)}, the circle with one point removed.

5. Union-Find

Implement the Union-Find datastructure/algorithm in Python.