What makes a space 'locally Euclidean' Euclidean?

Question

I come from a Physics background so I realise if my knowledge is lacking in this. I'm following an introduction to manifolds and to start, I found this definition of a 'locally Euclidean' topological space from Wikipedia:

A topological space $X$ is called locally Euclidean if there is a non-negative integer $n$ such that every point in $X$ has a neighborhood which is homeomorphic to real n-space $\mathbb{R}^n$

What defines whether a space is 'Euclidean'? To me, I would associate a Euclidean space as a space where we can define a length element $ds^2$ as the sum of the squares of cartesian coordinates eg. $ds^2 = dx^2 + dy^2 + dz^2+...$. This doesn't seem to fit in this context because no where in the definition of a manifold include a notion of distance. Also, I don't see how having a homeomorphism relates to this notion of distance.

So why does the existence of a homeomorphism have anything to do with being Euclidean?

Answer 1

This is a really valid complaint about overuse of an adjective. Like most people, you probably first encountered the word "Euclidean" in the context of geometry, where probably you studied Euclid's fifth postulate and how it characterizes "flat space," and we could have negative or positively curved space without it.

In this case, we are only speaking of topology. For example, the topology of $\mathbb{R}^2$ is the same as that of the hyperbolic plane (they are homeomorphic, the homeomorphism just doesn't preserve distance). So when we say a topological space is locally (or globally) Euclidean, we just means it looks like Euclidean space topologically.

Answer 2

The topological space $\mathbb{R}^n$ (and any space homeomorphic to it) is also called ''Euclidean $n$-space'' and a bunch of other things containing the adjective ''Euclidean''.

Answer 3

"The" Euclidean space of dimension $n\in\mathbb{N}$ is, by definition, $\mathbb{R}^n$, endowed with the topology induced by the usual Euclidean metric, i.e. $$ d(x,y) = \sqrt{ \sum_{j=1}^{n} |x_j-y_j|^2 }, $$ as well as the standard vector space structure over $\mathbb{R}$. Note that a Euclidean space carries a lot more structure than just the topological structure (it has a metric, there is a vector space structure running around, there is an inner product (which actually induces a norm from which the metric can be recovered), etc).

Note that, while this has historical links to the geometry of Euclid, the modern definition is largely unrelated, and has been abstracted away from classical Greek geometry—the modern notion of "the Euclidean space" typically involves all of the structure described above (especially the metric and inner product, which can be used to measure distance and angle, but also the other structures). On the other hand, "a Euclidean space" may refer to any space which has some specific number of these features, and "a locally Euclidean space" only has the (local) topological properties described.

When talking about a locally Euclidean space, as defined in the question, most of this structure is forgotten. Only the topological structure matters. So a space if locally Euclidean if each point has some neighborhood which is topologically equivalent to some Euclidean space. The topology on that original space needn't come from a metric, and that space needn't have any deeper structure—we only care about the topology at this level.

Answer 4

In mathematics the term "Euclidean space" typically refers to $\Bbb{R}^n$ together with its standard topological, metric and algebraic structure.

You've correctly noticed that term "locally Euclidean" does not involve metric at all, and as such is a purely topological term. This leads to the definition of "topological manifold". But if we want more structure on it, e.g. if we want to do calculus on a manifold (which os often desired) then this is not enough. And you have to introduce the notion of being differentiable, which deep down already is closely related to the particular choice of metric and algebraic structure on $\Bbb{R}^n$.