Euclidean topology

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{p_1^2 + \cdots + p_n^2}.

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{\left(p_1 - q_1\right)^2 + \left(p_2 - q_2\right)^2 + \cdots + \left(p_i - q_i\right)^2 + \cdots + \left(p_n - q_n\right)^2}.

In any metric space, the open balls form a base for a topology on that space. The Euclidean topology on \R^n is the topology generated 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{x \in \R^n : d(p,x) 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{A} \cap B = A \cap \overline{B} = \varnothing, where \overline{A} 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.

References

References

1. [[Metric space. Metric space#Open and closed sets.2C topology and convergence]]
2. Steen, L. A.. (1995). "[[Counterexamples in Topology]]". Dover.