Neighborhoods in Standard Discrete Metric Space

Theorem
Let $M = \struct {A, d}$ be a metric space where $d$ is the standard discrete metric.

Let $a \in A$.

Then $\set a$ is a neighborhood of $a$ which forms a basis for the system of neighborhoods of $a$.

Proof
By definition of the standard discrete metric:


 * $\map d {x, y} = \begin {cases}

0 & : x = y \\ 1 & : x \ne y \end {cases}$

Let $\epsilon \in \R_{>0}$ such that $\epsilon < 1$.

Then:

So by definition $\set a$ is a neighborhood of $a$.

Let $\NN_a$ be the system of neighborhoods of $a$.

Let $N \in \NN_a$ be a neighborhood of $a$.

Then:

Hence the result by definition of basis of a system of neighborhoods.