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# Euclidean space

## 1 Algebra

vectors = points

Algebra of vectors

Properties

## 2 Geometry

distance

$|| \cdot ||$ is the Euclidean norm.

## 3 Topology

The standard base of the topology is the collection of all "open" balls in $X={\bf R}^n$: $$\gamma_b = \{B(a, \delta ) \colon a \in {\bf R}^n, \delta > 0 \}, {\rm \hspace{3pt} where \hspace{3pt}} B(p,d) = \{u \colon |u-p| < d \}$$

is a basis. Rectangles, ellipses, etc also work (in ${\bf R}^2$).

Any two distinct points have neighborhoods that don't intersect. Indeed, if $x,y$ are these points in $X$, we define: $$U = B(x,r) {\rm \hspace{3pt} and \hspace{3pt}} V = B(y,r), {\rm \hspace{3pt} where \hspace{3pt}} r = \frac{||x-y||}{2}.$$

In other words, ${\bf R}^n$ is Hausdorff.

The topology of ${\bf R}^n$ can be understood as the product topology of $n$ copies of ${\bf R}$: $${\bf R}^n = {\bf R} \times \ldots \times {\bf R}.$$

Theorem. A closed bounded subset of a Euclidean space is compact.

Unbounded subsets in the Euclidean space aren't compact. For example, ${\bf R}$ can't even be covered by a finite collection of finite intervals.