Formal definition of euclidean space

Question

Is there an agreed upon, precise definition of the euclidean space? Although I have searched in may places, I can't find anything that is rigorous. From what I have read, I have thought of defining it like this:

Definition: The euclidean space of $n$ dimensions, $E^n$, is defined as the topology generated by the basis ($R^{n},d$), where $R^{n}$ is the set (Not the cartesian product of the standard real line topology) and $d$ is the Euclidean metric $d(x,y) = \Sigma^{n}_{i=1}(x^{2}_{i}-y^{2}_{i})$ (where $x = (x_{1},\dots ,x_{n})$ and $y=((y_{1},\dots ,y_{n})$).

Would the definition above be accurate?

Similarly, would it be accurate to define the $n$ sphere, $S^{n}$ (as a topological space) as the subset topology of $\{p \in R^{n} | d(x,p) = a\}$ (Where $a \in R^{+}$ and $x \in R^{n}$) inherited from the euclidean topology?

Answer 1

Ray Bowen, in his Introduction to Vectors and Tensors, Vol 2, section 43, defines a "Euclidean Point Space":

Consider an inner product space $V$ and a set $E$. The set $E$ is a Euclidean point space if there exists a function $f\colon E \times E \to V$ such that:

(a) $f(x, y) = f(x, z) + f(z, y)$, for $x, y, z\in E$ and

(b) For every $x\in E$ and $v\in V$ there exists a unique element $y\in E$ such that $f(x, y) = v$.

The elements of $E$ are called points, and the inner product space $V$ is called the translation space. We say that $f(x, y)$ is the vector determined by the end point $x$ and the initial point $y$.

Answer 2

The formal definition is probably something close to this.

Euclidean $n$-space, sometimes called Cartesian space or simply $n$-space, is the space of all $n$-tuples of real numbers, ($x_1, x_2, ..., x_n$). Such $n$-tuples are sometimes called points, although other nomenclature may be used (see below). The totality of $n$-space is commonly denoted $\mathbb R^n$, although older literature uses the symbol $\mathbb E^n$ (or actually, its non-doublestruck variant $E^n$; O'Neill 1966, p. 3).