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 MÃ¶bius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ...
For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ...
In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. ...
Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...
A synonym for ball (in geometry or topology, and in any dimension) is disk (or disc Geometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. ...
In mathematics, informally speaking, a limit point (or cluster point) of a set S in a topological space X is a point x in X that can be approximated by points of S other than x as well as one pleases. ...
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. In general a discrete set of Euclidean space is closed, but it is not open. See also discrete space. In mathematics the term countable set is used to describe the size of a set, e. ...
In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ...
A closed set with no isolated point is called a perfect set. In mathematics a derived set is a construction in point-set topology that consists of taking the set of limit points of a given subset S of a topological space X. The derived set of S is usually denoted by S′. A subset S of a topological space X...
## Examples
Topological spaces in the following examples are considered as subspaces of the real line. In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology (or the relative topology, or the induced topology). ...
In mathematics, the real line is simply the set of real numbers. ...
- For the set , the point 0 is an isolated point.
- For the set , 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. In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...
## See also In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ...
In topology and related branches of mathematics, an action of a group G on a topological space X is called properly discontinuous if every element of X has a neighborhood that moves outside itself under the action of any group element but the trivial element. ...
In mathematics, a free regular set is a subset of a topological space that is acted upon disjointly under a given group action. ...
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