Arbitrarily close

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Informally, a set $S$ contains points arbitrarily close to some point $p$ if, for any positive distance $\varepsilon$, there are members of $S$ which are less than a distance $\varepsilon$ from $p$.

More formally, in a metric space $M$ (such as the Euclidean space $\mathbb{R}^n$ or just the real numbers) with distance function $d$, a set $S \subset M$ is said to contain members arbitrarily close to some point $p \in M$ if for all $\varepsilon > 0$ there exists some $x \in S$ such that $0 < d(x, p) < \varepsilon$.


  • In the particular case of the real numbers with the usual distance $d(x, y) = |x - y|$, the set $S = \left\{\frac{1}{n} \mid n \in \mathbb{Z}_{> 0}\right\}$ contains points arbitrarily close to 0.
  • The set of real numbers contains points arbitarily close to any given real number. This is also true of the rational numbers, but it is not true of the integers. We can see this last fact is true because (for example) there is no integer at distance $0.1$ or less from $\pi$. (A set $S$ which contains points arbitrarily close to every point is said to be dense.)

See also

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