Let $n\in\mathbb{N}$. Consider the set $\mathbb{R}^n$ appearing in \reftext{def:open-subset-euclidean-space-2026a}{the definition of open subsets of Euclidean space}. For points

in $\mathbb{R}^n$, define

where $\sqrt{\ \cdot\ }$ denotes the nonnegative square root from \ref{thm:nonnegative-real-has-unique-square-root-2026a}. This is a function on the \reftext{def:cartesian-product-sets-2026a}{Cartesian product} $\mathbb{R}^n\times\mathbb{R}^n$ with values in $\mathbb{R}$, and it is called the Euclidean distance on $\mathbb{R}^n$.