TheInfoList

In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on Euclidean n-space $\mathbb^n$ by the Euclidean metric. 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 $\mathbb^n$ is then simply the topology ''generated'' by these balls. In other words, the open sets of the Euclidean topology on $\mathbb^n$ are given by (arbitrary) unions of the open balls $B_\left(p\right)$ defined as $B_\left(p\right) := \$, for all $r > 0$ and all $p \in \mathbb^n$, where $d$ is the Euclidean metric.

Properties

* The real line, with this topology, is a T5 space. Given two subsets, say ''A'' and ''B'', of R with , where ''A'' denotes the closure of ''A'', there exist open sets ''SA'' and ''SB'' with and such that

References

{{Reflist Category:Topology