Discrete Set/Examples/Natural Numbers
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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