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Concept
Euclidean topology
Summary
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on -dimensional Euclidean space by the Euclidean metric.
The Euclidean norm on is the non-negative function defined by
Like all norms, it induces a canonical metric defined by The metric induced by the Euclidean norm is called the Euclidean metric or the Euclidean distance and the distance between points and is
In any metric space, the open balls form a base for a topology on that space.
The Euclidean topology on is the topology by these balls.
In other words, the open sets of the Euclidean topology on are given by (arbitrary) unions of the open balls defined as for all real and all where is the Euclidean metric.
When endowed with this topology, the real line is a T5 space.
Official source
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