If and are two points on the plane, their Euclidean distance is given by
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 and , then formula 1 can be generalized to by defining the Euclidean distance from to as
Notice that this distance coincides with absolute value when . 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 .
|Date of creation||2013-03-22 12:08:21|
|Last modified on||2013-03-22 12:08:21|
|Last modified by||drini (3)|