Last modified on 28 November 2009, at 14:18

Real Analysis/Open and Closed Sets

TerminologyEdit

The open ball in a metric space (X,d) with radius  \epsilon centered at a, is denoted  B(a, \epsilon) . Formally  B(a, \epsilon) = \{x \in X: d(a,x) < \epsilon\}

DefinitionEdit

Let  (X,d) be a metric space. We say a set  A \subset X is open if for every  x \in A \text{ } \exists \epsilon > 0 such that  B(x,\epsilon) \subset A .

We say a set B \subset X is closed if X\backslash B is open.