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Limit point

From AoPSWiki

Given a topological space $X$ and a subset $S$ of $X$ , an element $x$ of $X$ is called a limit point of $S$ if every neighborhood of $x$ contains some element of $S$ other than $x$ .

When $X$ is a metric space, it follows that every neighborhood of $x$ must contain infinitely many elements of $S$ . A point $x$ such that each neighborhood of $x$ contains uncountably many elements of $S$ is called a condensation point of $S$ .

Examples

Let $X = \mathbb{R}$ be the space of real numbers (with the usual topology) and let $S = \{\frac{1}{n} \mid n \in \mathbb{Z}_{> 0}\}$ , that is the set of reciprocals of the positive integers. Then $0$ is the unique limit point of $S$ .

Let $X = \mathbb{R}$ and $S =\mathbb{Q}$ be the set of rational numbers. Then every point of $X$ is a limit point of $S$ . Equivalently, we may say that $\mathbb{Q}$ is dense in $\mathbb{R}$ .

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