In topology, a branch of mathematics, a point x of a set S is called an isolated point, if there exists a neighborhood of x not containing other points of S. In particular, in a 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 an accumulation point. A set which is made up only of isolated points is called a discrete set. A discrete subset of Euclidean space is countable; however, a set can be countable but not discrete, e.g. the rational numbers. See also discrete space. A closed set with no isolated point is called a perfect set. The number of isolated points is a topological invariant, i.e. if two topological spaces <math>X</math> and <math>Y</math> are homeomorphic, the number of isolated points in each is equal.
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 <math>{\mathbb N} = \{0, 1, 2, \ldots \}</math> of natural numbers is a discrete set.
See also
External links
- http://www.cool-rr.com/protein.htm Rigorous proof of isolated points' countability.
