If $u=(x_{1},y_{1})$ and $v=(x_{2},y_{2})$ are two points on the plane, their Euclidean distance is given by

Geometrically, it’s the length of the segment joining $u$ and $v$, and also the norm of the difference vector (considering $\mathbbmss{R}^{n}$ as vector space).

This distance induces a metric (and therefore a topology) on $\mathbbmss{R}^{2}$, called Euclidean metric (on $\mathbbmss{R}^{2}$) or standard metric (on $\mathbbmss{R}^{2})$. The topology so induced is called standard topology or usual topology on $\mathbbmss{R}^{2}$ 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 1 can be generalized to $\mathbbmss{R}^{n}$ by defining the Euclidean distance from $a$ to $b$ as

Notice that this distance coincides with absolute value when $n=1$. Euclidean distance on $\mathbbmss{R}^{n}$ is also a metric (Euclidean or standard metric), and therefore we can give $\mathbbmss{R}^{n}$ a topology, which is called the standard (canonical, usual, etc) topology of $\mathbbmss{R}^{n}$. The resulting (topological and vectorial) space is known as Euclidean space.

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