Coordinate chart

[Rasalhague]

The reason the distinction is made is that the point 0 is "marked" in a vector space, but it is not in Euclidean space.

That's a distinction between a vector space and an affine space, but I often find the term "Euclidean space" defined as a vector space, as here

http://mathworld.wolfram.com/EuclideanSpace.html

So I'm interested in what the ideal way of defining "Euclidean space" and "Rⁿ" might be, and also in the (apparently varied) ways that they're defined in practice, so that I can understand mathematical texts. I think your point is that it's best to regard Euclidean space as an affine space.

For our purposes, we don't need the vector space structure, so it's superfluous.

The definitions that I've read of an affine space all include the concept of a vector space. For example, I've seen affine space defined as the 3-tuple (S,V,f) where S is a nonempty set (whose elements are called the points), V is a vector space, and f is a function, +, relating points in S and vectors of V, such that

1. (p + u) + v = p + (u + v).

2. For all q in S, there exists a unique vector v in V such that q = p + v.

In addition, it's confusing, since we need to be able to identify "vectors" with differing base points, which we certainly can't do in the vector space context.

You mean bound vectors? That reminds me... Does this line from the article Euclidean vector seem the wrong way round to you: "This more general type of spatial vector is the subject of vector spaces (for bound vectors) and affine spaces (for free vectors)"?

 

[Rasalhague]

Suppose I choose the trivial topology (indiscrete topology, coarsest topology), consisting of the empty set and the improper subset, Rⁿ itself, and let Rⁿ denote the set of all real n-tuples. Does Wolfram Mathworld's definition of a manifold still apply: "A manifold is a topological space that is locally Euclidean (i.e., around every point, there is a neighborhood that is topologically the same as the open unit ball in Rⁿ)"? I thought an open ball was defined as the interior of a sphere, and a sphere as the set of points some constant distance from a specified point; but doesn't this require Rⁿ to be a metric space for there to be some notion of distance? Other definitions of manifold that I've seen also refer to open balls.

Also, regarding the term "topological space", can every set S be called a topological space with the trivial topology, and is the same set with a different topology regarded as a different topological space? (I'm thinking yes to both.)

Maybe the open ball in this definition would be replaced by an open set according to the second and more general definition here.

http://mathworld.wolfram.com/OpenSet.html

 

[wofsy]

A single open set that is homeomorphic to euclidean space - together with the homeomorphism - makes it a manifold .