Real Analysis/Open and Closed Sets

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Terminology[edit]

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\}

Definition[edit]

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.