Regard Q, the set of all rational numbers, as a metric space, with d(p, q) = |p − q|. Let E
be the set of all p ∈ Q such that 2 < p2 < 3. Show that E is closed and bounded in Q, but that E is not compact. Is E open in Q?
Definition of interior point, limit point, open set, closed set, and compact set.
The Attempt at a Solution
Obviously E is bounded, but I'm not convinced that E is closed. Take a point in Ec. Supposedly there is a neighborhood, about that point, containing only members of E. But that can't be true, since any neighborhood about that point will also contain irrational numbers (Right?).