Euclidean topology (${\tau }_{\Vert \cdot {\Vert }_2}$)

The Euclidean topology on ${\mathbb{R}}^n$ is the metric topology induced by the Euclidean norm. A subset is open exactly when each of its points has an open ball contained in it. It agrees with the finite product topology on $n$ copies of the real line: the inequalities $\Vert x{\Vert }_{\infty }\le \Vert x{\Vert }_2\le \sqrt{n}\Vert x{\Vert }_{\infty }$ put a coordinate box inside every sufficiently small ball and a ball inside every sufficiently small coordinate box. The case $n=0$ is the one-point topological space.