Euclidean topology

In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on $n$-dimensional Euclidean space $\R^n$ by the Euclidean metric.

Definition

The Euclidean norm on $\R^n$ is the non-negative function $\, \cdot\, : \R^n \to \R$ defined by
$\left\, \left(p_1, \ldots, p_n\right)\right\, ~:=~ \sqrt.$

Like all norms, it induces a canonical metric defined by $d(p, q) = \, p - q\, .$ The metric $d : \R^n \times \R^n \to \R$ induced by the Euclidean norm is called the Euclidean metric or the Euclidean distance and the distance between points $p = \left(p_1, \ldots, p_n\right)$ and $q = \left(q_1, \ldots, q_n\right)$ is
$d(p, q) ~=~ \, p - q\, ~=~ \sqrt.$

In any metric space, the open balls form a base for a topology on that space. Metric space#Open and closed sets.2C topology and convergence
The Euclidean topology on $\R^n$ is the topology by these balls.
In other words, the open sets of the Euclidean topology on $\R^n$ are given by (arbitrary) unions of the open balls $B_r(p)$ defined as $B_r(p) := \left\,$ for all real $r > 0$ and all $p \in \R^n,$ where $d$ is the Euclidean metric.

Properties

When endowed with this topology, the real line $\R$ is a T₅ space.
Given two subsets say $A$ and $B$ of $\R$ with $\overline \cap B = A \cap \overline = \varnothing,$ where $\overline$ denotes the closure of $A,$ there exist open sets $S_A$ and $S_B$ with $A \subseteq S_A$ and $B \subseteq S_B$ such that $S_A \cap S_B = \varnothing.$

See also

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References

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Topology