1. Suppose E ⊂ R, E ≠ ϕ, E bounded above but not below and E has the betweenness property. Show that either E = (-∞, b) for some b on E = (-∞, b] for some b.
2. Let E = {(x,y) ∈ R² : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}. Show that E is connected. Hint: Show that a separation of E would lead to a separation of [0, 1].
3. Show that if E is a closed subset of a compact set K, then E is compact.
4. a) Suppose X, Y are metric spaces, f X --> Y, f is continuous, E ⊂ Y, E is open. Show that f⁻¹ (E) is open, where f⁻¹(E) = {x ∈ X : f(x) ∈ E}
b) Let I = (0, 1) in R. Find a continuous f: R --> R such that f(I) = {f(x): x ∈ I} is not open.
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