Exercise 1186Topology › Topology of ℝL1Added 3 Nov 2026

Limit points

#limit points#accumulation#closure

Statement

Define a limit point of a set $A\subseteq\mathbb{R}$ and describe the limit points of $A=\{1/n:n\ge1\}$ .

Hint

$x$ is a limit point if every neighbourhood of $x$ meets $A\setminus\{x\}$ .

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Definition. $x$ is a limit point of $A$ if $\forall\varepsilon>0$ , $(x-\varepsilon,x+\varepsilon)\cap(A\setminus\{x\})\neq\emptyset$ .

Limit points of $A$ . Only $0$ : any $\varepsilon$ -neighbourhood of $0$ contains infinitely many $1/n$ ; other points are isolated.

Conclusion. $A'=\{0\}$ ; the closure is $\overline A=A\cup\{0\}$ .