Prove the closure of E in a Metric Space X is closed. (page 35) Rudin states: if p∈X and p∉E then p is neither a point of E nor a limit point of E.. Hence, p has a neighborhood which does not intersect E. (Great) The compliment of the closure of E is therefore open. WHY? I don't see it... BTW, I know there are different ways to proving this, but I want to understand the last line jump. Thanks.