Isolated point explained

In mathematics, a point is called an isolated point of a subset (in a topological space) if is an element of and there exists a neighborhood of that does not contain any other points of . This is equivalent to saying that the singleton is an open set in the topological space (considered as a subspace of). Another equivalent formulation is: an element of is an isolated point of if and only if it is not a limit point of .

If the space is a metric space, for example a Euclidean space, then an element of is an isolated point of if there exists an open ball around that contains only finitely many elements of .A point set that is made up only of isolated points is called a discrete set or discrete point set (see also discrete space).

Related notions

Any discrete subset of Euclidean space must be countable, since the isolation of each of its points together with the fact that rationals are dense in the reals means that the points of may be mapped injectively onto a set of points with rational coordinates, of which there are only countably many. However, not every countable set is discrete, of which the rational numbers under the usual Euclidean metric are the canonical example.

A set with no isolated point is said to be dense-in-itself (every neighbourhood of a point contains other points of the set). A closed set with no isolated point is called a perfect set (it contains all its limit points and no isolated points).

The number of isolated points is a topological invariant, i.e. if two topological spaces are homeomorphic, the number of isolated points in each is equal.

Examples

Standard examples

Topological spaces in the following three examples are considered as subspaces of the real line with the standard topology.

For the set

S=\{0\}\cup[1,2],

the point 0 is an isolated point.

For the set

S=\{0\}\cup\{1,\tfrac12,\tfrac13,...\},

each of the points is an isolated point, but is not an isolated point because there are other points in as close to as desired.

The set

\N=\{0,1,2,\ldots\}

of natural numbers is a discrete set.In the topological space

X=\{a,b\}

with topology

\tau=\{\emptyset,\{a\},X\},

the element is an isolated point, even though

belongs to the closure of

\{a\}

(and is therefore, in some sense, "close" to). Such a situation is not possible in a Hausdorff space.

The Morse lemma states that non-degenerate critical points of certain functions are isolated.

Two counter-intuitive examples

Consider the set of points in the real interval such that every digit of their binary representation fulfills the following conditions:

Either

x_i=0

x_i=1.

x_i=1

only for finitely many indices .

If denotes the largest index such that

x_m=1,

then

x_m-1=0.

x_i=1

and

i<m,

then exactly one of the following two conditions holds:

x_i-1=1

x_i+1=1.

Informally, these conditions means that every digit of the binary representation of

that equals 1 belongs to a pair ...0110..., except for ...010... at the very end.

Now, is an explicit set consisting entirely of isolated points but has the counter-intuitive property that its closure is an uncountable set.

Another set with the same properties can be obtained as follows. Let be the middle-thirds Cantor set, let

I_1,I_2,I_3,\ldots,I_k,\ldots

be the component intervals of

[0,1]-C

, and let be a set consisting of one point from each . Since each contains only one point from, every point of is an isolated point. However, if is any point in the Cantor set, then every neighborhood of contains at least one, and hence at least one point of . It follows that each point of the Cantor set lies in the closure of, and therefore has uncountable closure.

Isolated point explained

Related notions

Examples

Standard examples

Two counter-intuitive examples

See also