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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?

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    $\begingroup$ The product topology and metric topology are the same, in this case. $\endgroup$ Jan 5, 2019 at 15:09
  • $\begingroup$ All topologies generated by a norm on a finite-dimensional space over the real or complex numbers are in fact the same. $\endgroup$ Jan 5, 2019 at 15:11
  • $\begingroup$ So can I use the definition I have provided? (Sorry for the bluntness) $\endgroup$ Jan 5, 2019 at 15:14
  • $\begingroup$ I suspect that the short answer to your question is "no". There have been several similar questions before, including What is the difference between a Hilbert space and Euclidean space?, What really is the modern definition of Euclidean spaces?, Is the Euclidean plane equal to $\mathbb{R}^2$?. Personally I remain confused, so I'm just commenting, not answering! $\endgroup$ Jan 5, 2019 at 15:50
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    $\begingroup$ Following up on the comment of CalumGilhooley, this question invites ambiguous answers because it does not address the following key issue: What are the salient mathematical structures of Euclidean space? Simply its points, as the answer of @ItsJustAMeasureBro suggests? Its points together with its lines? Its points together with its metric, as the definition in your question suggests? Its points, lines, metric, angles, and angle measurement? All of the above together with its inner product, as one link in Calum's comment suggests? ......... $\endgroup$
    – Lee Mosher
    Jan 5, 2019 at 18:54

2 Answers 2

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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 $y$.

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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).

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  • $\begingroup$ Depending on the context, Euclidean Space usually doesn't fix an origin and so there is a distinction between points and translations between them. $\mathbb R^n$ can be used to stand for both the points and the translations, but that's sort of incidental. As an aside, I wouldn't personally trust mathworld for definitions; I've found pages with multiple inconsistent definitions, pages with definitions that seem to agree with no other internet source, etc. $\endgroup$
    – Mark S.
    Jan 8, 2019 at 11:03

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