general topology

• Are Singleton Sets in ℝ Both Open and Closed?
  In ℝ with the usual topology, a singleton {a} is closed but not open. Proofs of both facts, and why only ∅ and ℝ are both open and closed.
• Is Every Singleton Set Open? (No, Here's When It Is)
  Singleton sets are not open in general. In ℝ, {0} is closed but not open. A singleton {x} is open exactly when x is an isolated point, as in discrete spaces.
• Why Is Every Singleton Set Open in a Finite Metric Space?
  Proof that {x} is open in any finite metric space, using the minimum distance to the other points, and why this makes every subset open (the discrete topology).