For metric spaces $(M_1, d_1)$ and $(M_2, d_2)$, it is an exercise that the product topology on $M_1\times M_2$ is induced by the metric $d=d_1 + d_2$.

Do you know if this statement generalises to premetric spaces?

Here, we call $(M,\tilde{d})$ a premetric space if $M$ is a set and $\tilde{d}:M\times M\rightarrow[0,\infty)$ is such that $\tilde{d}(x,x)=0$ for all $x\in M$.

For metric spaces $(M_1, d_1)$ and $(M_2, d_2)$, it is an exercise that the product topology on $M_1\times M_2$ is induced by the metric $d((x_1, y_1), (x_2, y_2)) =d_1(x_1, x_2) + d_2(y_1, y_2)$.

Do you know if this statement generalises to premetric spaces?

Here, we call $(M,\tilde{d})$ a premetric space if $M$ is a set and $\tilde{d}:M\times M\rightarrow[0,\infty)$ is such that $\tilde{d}(x,x)=0$ for all $x\in M$.