The Topology of Euclidean Space#

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 real 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 \((\mathbb{R}^n, \operatorname{id})\), where \(\operatorname{id}\) 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

Theorem: Connectedness of Euclidean Space

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

Proof

TODO

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

Definition: Open Cuboid

An open cuboid in the Euclidean space \(\mathbb{R}^n\) is a subset \(C\) of \(\mathbb{R}^n\) which can be represented using \(n\) open intervals \(I_1, \cdots, I_n\) as

\[ C = \left\{ \begin{bmatrix}c_1 \\ \vdots \\ c_n \end{bmatrix} : c_1 \in I_1, \dotsc, c_n \in I_n \right\} \]

We call \(I_1, \dotsc, I_n\) the sides of \(C\).

Definition: Unit \(n\)-Sphere

The unit \(n\)-sphere is the topological subspace of the Euclidean space \(\mathbb{R}^n\) defined by the subset

\[ \{\mathbf{p} \in \mathbb{R}^{n+1} : ||\mathbf{p}|| = 1\} \]

Notation

\[ \mathbb{S}^n \]

Definition: Closed Upper Half-Space

The closed \(n\)-dimensional upper half-space is the topological subspace of the Euclidean space \(\mathbb{R}^n\) defined by the subset

\[ \left\{ \begin{bmatrix}x_1 \\ \vdots \\ x_n \end{bmatrix} \in \mathbb{R}^n \mid x_n \ge 0 \right\} \]

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 Euclidean distance from \(p\) is less than \(r\). It is essentially an \(n\)-dimensional sphere which is centered at \(p\) and has radius \(r\).

The open subsets of the Euclidean space \(\mathbb{R}^n\) are thus:
- All open balls;
- All unions of arbitrary collections of open balls;
- All intersections of finite collections of open balls.

The Topology of the Real Number Line#

The real number line is the Euclidean space \(\mathbb{R}\).

Definition: Intervals

An open interval is a subset \((a;b)\) of the real number line which has the form \((a;b) \overset{\text{def}}{=} \{x \in \mathbb{R} \mid a \lt x \lt b\}\) for some \(a \in \mathbb{R} \cup \{-\infty\}\) and \(b \in \mathbb{R} \cup \{+\infty\}\).

A closed interval is a subset \([a;b]\) of the real number line which has the form \([a;b] \overset{\text{def}}{=} \{x \in \mathbb{R} \mid a \le x \le b\}\) for some \(a, b \in \mathbb{R}\).

A semi-open interval (or semi-closed interval) is a subset of the real number line which has one of the following forms:

\((a;b] \overset{\text{def}}{=} \{x \in \mathbb{R} \mid a \lt x \le b\}\) for some \(a \in \mathbb{R} \cup \{-\infty\}\) and \(b \in \mathbb{R}\);
\([a;b) \overset{\text{def}}{=} \{x \in \mathbb{R} \mid a \le x \lt b\}\) for some \(a \in \mathbb{R}\) and \(b \in \mathbb{R} \cup \{+\infty\}\).

Theorem: Connectedness on the Real Line

A

The intervals are precisely the connected subsets of \(\mathbb{R}\).

Proof

TODO

Theorem

Every closed interval of the real number line \(\mathbb{R}\) is compact.

Proof

TODO