Fundamentals of Topology

Abstract

• We say A ⊂ ℝ is open if for every x ∈ A there exists ε > 0 such that (x − ε, x + ε) ⊆ A. A is closed if its complement A ^c is open. Similarly a set A in a metric space (M,d) is called open if for each x ∈ A, there exists an ε > 0 such that B(x;ε) ⊂ A. Here,

A linguist would be shocked to learn that if a set is not closed this does not mean that it is open, or again that “E is dense in E” does not mean the same thing as “E is dense in itself.”

John Edensor Littlewood (1885–1977)