Discrete Set/Examples/Natural Numbers

Example of Discrete Set

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 $T$.

Proof

Let $x \in \N$ be an arbitrary natural number.

Let $\map B {x; \dfrac 1 2} \subseteq \R$ be the open ball of radius $\dfrac 1 2$ on $\R$ whose center is $x$.

Then $x$ is the only natural number in $\map B {x; \dfrac 1 2}$.

Hence by definition $x$ is isolated in $\N$.

As $x$ is arbitrary, the result follows.

$\blacksquare$

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