Euclidean Space

The Euclidean Topology

Definition: The Euclidean Topology

The Euclidean topology on the real vector space ${\mathbb{R}}^n$ is the metric topology induced on it by the Euclidean metric.

The vector space ${\mathbb{R}}^n$ equipped with its Euclidean topology is known as the $n$-dimensional Euclidean space.

Theorem: Euclidean Space is a Smooth Manifold

The atlas whose only chart is $({\mathbb{R}}^n,\mathrm{id})$, where $\mathrm{id}$ is the identity function on the Euclidean space ${\mathbb{R}}^n$, is a smooth structure on ${\mathbb{R}}^n$ and naturally makes it a smooth manifold.

PROOF

TODO

Open Sets

Intuition: Open Balls in Euclidean Space

Let $p\in {\mathbb{R}}^n$ be an element of the Euclidean space ${\mathbb{R}}^n$.

An open ball $B_r(p)$ of radius $r$ around $p$ contains all $x\in {\mathbb{R}}^n$ whose distance from $p$ is less than $r$. It is essentially an $n$-dimensional sphere which is centred at $p$ and has radius $r$.

Tip: Open Sets in Euclidean Space

The open subsets of the Euclidean space ${\mathbb{R}}^n$ are precisely the following:

• All open balls;
• All unions of arbitrary collections of open balls;
• All intersections of finite collections of open balls.

Connectedness

Theorem: Connectedness of Euclidean Space

The Euclidean space ${\mathbb{R}}^n$ is connected.

PROOF

TODO

Theorem: Connectedness on the Real Line

All intervals and rays in the Euclidean space $\mathbb{R}$ of the real line are connected.

PROOF

TODO

Compactness

Heine-Borel Theorem: Compactness in Euclidean Space

A subspace of the Euclidean space ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded.

PROOF

TODO

THEOREM

Every closed interval of the Euclidean space $\mathbb{R}$ is compact.

PROOF

TODO