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Euclidean distance
If and are two points on the plane, their Euclidean distance is given by
| (1) |
Geometrically, it’s the length of the segment joining and , 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 and , then formula 1 can be generalized to by defining the Euclidean distance from to as
| (2) |
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 .
Mathematics Subject Classification
53A99 None of the above, but in MSC2010 section 53Axx54E35 Metric spaces, metrizability
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