If $u=(x_1,y_1)$ and $v=(x_2,y_2)$ are two points on the plane, their Euclidean distance is given by \begin{equation}\label{equno} \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}. \end{equation}Geometrically, it's the length of the segment joining $u$ and $v$ , and also the norm of the difference vector (considering as vector space).

This distance induces a metric (and therefore a topology) on , called Euclidean metric (on ) or standard metric (on . The topology so induced is called standard topology or usual topology on and one basis can be obtained considering the set of all the open balls.

If $a=(x_1,x_2,\ldots,x_n)$ and $b=(y_1,y_2,\ldots,y_n)$ , then formula can be generalized to by defining the Euclidean distance from $a$ to $b$ as \begin{equation}d(a,b)=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2+\cdots+(x_n-y_n)^2}.\end{equation} Notice that this distance coincides with absolute value when $n=1$ . Euclidean distance on is also a metric (Euclidean or standard metric), and therefore we can give a topology, which is called the standard (canonical, usual, etc) topology of . The resulting (topological and vectorial) space is known as Euclidean space.

This can also be done for since as set and thus the metric on is the same given to , and in general, gets the same metric as $R^{2n}$ .