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Euclidean distance
If $u=(x_1,y_1)$ and $v=(x_2,y_2)$ are two points on the plane, their Euclidean distance is given by
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 1 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 $R^{2n}$ .