Isolated point
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In topology, a point x of a set S is called an isolated point, if there exists a neighbourhood of x not containing other points of S. In particular, in an Euclidean space (or in a metric space), x is an isolated point of S, if one can find an open ball around x which contains no other points of S.
A set which is made up only of isolated points is called a discrete set.
Examples
- For the set <math>S=\{0\}\cup [1, 2]<math>, the point 0 is an isolated point.
- For the set <math>S=\{0\}\cup \{1, 1/2, 1/3, \dots \}<math>, each of the points 1/k is an isolated point, but 0 is not an isolated point because there are other points in S as close to 0 as desired.
- The set N={0, 1, 2, ...} of natural numbers is a discrete set.