P-Product Metric Induces Product Topology

Theorem
Let $M_A = \left({A, d_A}\right)$ and $M_B = \left({B, d_B}\right)$ be metric spaces.

Let $\tau_A$ and $\tau_B$ be the topologies on $A$ and $B$ induced by $d_A$ and $d_B$, respectively.

Let $p \ge 1$ be an extended real number.

Let $M = \left({A \times B, d}\right)$ be the $p$-product of $M_A$ and $M_B$.

Then $M$ is a metric space.

Let $\tau$ be the topology on $A \times B$ induced by $d$.

Then $\left({A \times B, \tau}\right)$ is the topological product of $\left({A, \tau_A}\right)$ and $\left({B, \tau_B}\right)$.