Definition:Discrete Set

Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $H \subseteq S$ be a subset of $S$.

Then $H$ is a discrete set if and only if every point of $H$ is an isolated point of $H$.

Examples

Natural Numbers

Let $T = \struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

The natural numbers $\N$ form a discrete set within $T$.

Rational Numbers

Let $T = \struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

The rational numbers $\Q$ do not form a discrete set within $T$.

Also see

• Definition:Dense Set

• Results about discrete sets can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): discrete set
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): discrete set