Closure

Topics: Calculus - Topology

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(definition)

We define the closure of a set $A$ as:

$$\overline{A}=A\cup A_c(A)$$

…where $A_c(A)$ is the set of all accumulation points of $A$.

Examples

• $\overline{(0,1)}=[0,1]$
• $\overline{\mathbb{Q}}=\mathbb{R}$
• $\overline{\mathbb{I}}=\mathbb{R}$
• $\overline{\{\frac{1}{n}\mid n\in \mathbb{N}\}}=\{\frac{1}{n}\mid n\in \mathbb{N}\}\cup \{0\}$