I am confused by some definitions. Forgive the looseness of my language.
A Cartesian space is basically a space of points that can be represented by n-tuples ( and other things, but I won't go into specifics). Open sets are denoted by $|x^i_{0}-x^i|<{\epsilon}$. It is denoted by $R^n$ (according to Dubrovin's book).
Euclidean spaces are Cartesian spaces where there exists co-ordinate systems in which the metric can be represented by the Kronecker delta. Its open sets can be represented as open balls with the usual definition of 'balls'.
Riemannian spaces are Cartesian spaces which have a positive definite quadratic form on its tangent vectors. Euclidean spaces are special cases of Riemannian spaces. So I imagine that open sets are determined by the riemannian metric? or are they determined by euclidean balls?
Now manifolds are spaces, whose subsets can be identified with Euclidean spaces. But such charts can be equipped with additional Riemannian metric.
So I am confused. Does that mean that the charts on manifolds have 2 'metrics', a fundamental one determined via the identification with a region of euclidean space and second additional riemannian metric?
Also open sets on a manifold may be defined by identifying them with the open sets in Euclidean space. So then how does the additional riemannian metric help determine open sets on the manifold?
I am totally confused about the hierarchy of these spaces, if Cartesian is on the top and Riemannian is below and Euclidean even lower than that, than why are manifolds defined first by identification with Euclidean spaces and then given additional structure of a Riemannian metric? Isn't this the inversion of the given hierarchy?
When answering please don't be ambiguous with the words and notations, for it will only compound my confusion.
Every smooth $n$-dimensional manifold $M$ comes equipped with an atlas of charts $\phi_\alpha: U_\alpha\to R^n$. Of course, on $R^n$ we have the standard (flat) Riemannian metric, usually denoted $\delta_{ij}$. However, the transition maps $\phi_\beta\circ \phi_\alpha^{-1}$ (typically) do not preserve these metrics. Another way to put is that $\phi_\alpha^*(\delta_{ij})\ne \phi_\beta^*(\delta_{ij})$ on the intersection $U_\alpha\cap U_\beta$. Therefore, one (typically) cannot use the metrics $\delta_{ij}$ to define a Riemannian metric on the entire $M$. For concreteness, you can work out the example of $M=S^2$ and two charts given by stereographic projections. The transition map will be an inversion which does not preserve the flat metric on $R^2$.
In fact, with this definition of a smooth manifold, it is unclear that it admits any Riemannian metric whatsoever; one can, however, give an argument that it does, say, by embedding the entire $M$ in some $R^N$, $f: M\to R^N$, and then taking the pull-back Riemannian metric $f^*(\delta_{ij})$. (There are other constructions, of course.)