Informally, a set contains points arbitrarily close to some point if, for any positive distance , there are members of which are less than a distance from .
More formally, in a metric space (such as the Euclidean space or just the real numbers) with distance function , a set is said to contain members arbitrarily close to some point if for all there exists some such that .
- In the particular case of the real numbers with the usual distance , the set 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 or less from . (A set which contains points arbitrarily close to every point is said to be dense.)
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