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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.## Examples

## See also

## External links

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 $X$ and $Y$ are homeomorphic, the number of isolated points in each is equal.

Topological spaces in the following examples are considered as subspaces of the real line.

- For the set $S=\{0\}cup\; [1,\; 2]$, the point 0 is an isolated point.
- For the set $S=\{0\}cup\; \{1,\; 1/2,\; 1/3,\; dots\; \}$, 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 $\{mathbb\; N\}\; =\; \{0,\; 1,\; 2,\; ldots\; \}$ of natural numbers is a discrete set.

- http://www.cool-rr.com/protein.htm Rigorous proof of isolated points' countability.

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Last updated on Tuesday September 23, 2008 at 17:49:20 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Tuesday September 23, 2008 at 17:49:20 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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