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?
Best Answer
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.