18.099: Problem Set 4

Due: Tuesday, October 7.

1. Let F ⊂ ℝ be a closed subset and suppose that α = sup F exists. Show that α ∈ F. (Hint: If α ∉ F then α ∈ ℝ ∖ F.)

2. Let X be a metric space, let F ⊂ X be a closed subset, and let K ⊂ X be a compact subset. Show that F ∩ K ⊂ X is a compact subset.

3. Let X be a set. The discrete metric on X is given by the map

d: X × X → ℝ

that takes (p, q) to 0 and 1, respectively, as p = q and p ≠ q. Show that every subset E ⊂ X is open. Conclude that every subset F ⊂ X is closed. Finally, which subsets K ⊂ X are compact?