Isolated point
From Free net encyclopedia
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. Equivalently, a point x is not isolated if and only if x is a limit point.
A set which is made up only of isolated points is called a discrete set, e.g., finite set. A discrete subset of Euclidean space is countable; however, a set can be countable but not discrete, e.g. the rational numbers.
A closed set with no isolated point is called a perfect set.
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Examples
Topological spaces in the following examples are considered as subspaces of the real line.
- 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.
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