Definition:Discrete Set
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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
- 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