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.

Examples

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.

See also

• Freely discontinuous
• Free regular set

External links

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