Complex Analysis Problems

1. Suppose that E is any set with ordering such that all non empty subsets of E which are bounded above have a L.U.B.
Show that If A ⊂ E, A ≠ ∅ and A is bounded below, then A has a greatest lower bound. (Hint: Consider G = The set of all lower bounds of A]

2. a) Let I be a set, X a metric space, and that for each i ∈ I we have an open subset Vᵢ of X, let V = U Vᵢ (def {x: x ∈ V for at least one i ∈ I}). Show that V is open
b) Suppose Vᵢ is open in X, i = 1, ... , n and let W = ∩ Vᵢ. Show that W is open.
c) Find open intervals J₁, J₂, J₃ ... in R, such that ∩ Jᵢ is not open.

3. In a metric space X, let E be a subset of X, V = Interior(E).
a) Show that V is open.
b) Show that V is the "largest open subset of E", more precisely: IF W ⊂ E, W is open, then W ⊂ V.

4. Define the metric d on X = {(x,y) ∈ R²: 0 ≤ x ≤ 1, 0 ≤ y ≤ 1} by d((x, y), (x', y')) = {1, if y ≠ y' and |x - x'|, if y = y'} (You can assume d is in fact a metric)
In each case, find in (X, d), the interior and closure of E.
a) E = {(1/n, y): n ∈ N, 0 ≤ y ≤ 1}
b) E = {(x, 1/n): 0 ≤ x ≤ 1, n ∈ N}

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