# Complex Analysis Problems

## Question

1. Define d: R² -> R by d((x,y), (x', y')) = (|x - x'|, |y - y'|)
a) Show d is a metric
b) Draw a picture showing N₁((1,1)) in R²:
i) Using the metric d
ii) Using the taxicab metric

2. a) Show, in any metric space X:
i) ϕ is closed, X is closed
ii) Any intersection of closed sets is closed
iii) Any finite union of closed sets is closed
b) Give an example of closed intervals F₁, F₂, ... in R such that Uₙ ₌ ₁ to infinity Fₙ is not closed.

3. Define d: R² -> R by d((x,y), (x', y')) = 1, if x ≠ x' and min(|y-y'|, 1|) if x = x'.
State (explaining your answer) whether each of the following sets is open, closed, or neither:
a) E₁ = {(x,y): 0 < x < 1, 0 < y < 1}
b) E₂ = {(x,y): 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}
c) E₃ = {(x,y): 0 < x < 1, 0 ≤ y ≤ 1}
d) E₄ = {(x,y): 0 ≤ x ≤ 1, 0 < y < 1}

4. Let (X, d) be a metric space, E ⊂ X. Show G(Int(E)) = G(E)

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