Metric is Continous Mapping

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $M = \struct {A, d}$ be a metric space.

Consider the distance function:

$d: A \times A \to \R$

Then $\R$ is a continuous function.


Proof

Let $\epsilon > 0$.

Let $\tuple {x_1, x_2} \in A \times A$.

Let $\delta = \dfrac \epsilon 2$.

Then $U = \map {B_\delta} {x_1} \times \map {B_\delta} {x_2}$ is a neighborhood of $\tuple {x_1, x_2}$ in $X \times X$ such that:

$d \sqbrk U \subseteq \openint {\map d {x_1, x_2} - \epsilon} {\map d {x_1, x_2} + \epsilon}$

Therefore $d: X \times X \to \mathbb R$ is a continuous function.

$\blacksquare$