Is there a space that does not include or need a coordinate system? A coordinate system (such as cartesian and polar coordinate systems) is expected in a Euclidean space. However, is there a space that does not include or need a coordinate system?
I read that a Vector space does not need a coordinate system although a coordinate system can be defined if needed.

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*Is there any other space that does not need/include a coordinate system?

*Is there any space that does not accept a coordinate system?

 A: There are mathematicians, such as Pavel Grinfeld, who advocate the idea that space is indifferent to coordinate systems. Motion and geometry in space, says he, do not care what coordinate system we use to describe it, and indeed you find that there are several useful choices of coordinates systems, which serve as tools: if you want to analyse a geometric or physical problem in space which strongly depends on the distance from a certain point, you may want to use polar coordinates; if you have motion on the surface of a sphere, or a cylinder, you may want to use spherical or cylindrical coordinates, etc.
That said, your first question is not a proper mathematical question. There is no notion such as "a space needs this" or a "space needs that". Space does not need anything. It is us who need coordinate systems to describe space.
As for the second question, it depends on your definition of a "space". If you mean a topological space, then probably the answer is probably "yes": general topological spaces do not accept a natural definition of coordinates. If you mean a "vector space", then since every vector space has an algebraic basis, it is -- at least in principle -- possible to associate a system of coordinates in every vector space. A basis in a vector space is a choice of a coordinate system. However, in its most abstract form, it is probably not very useful. The most useful vector spaces are those which have additional structure -- such as Hilbert spaces, Banach spaces, and in general, Topological vector spaces. All of these of course have a choice of coordinate systems, but in order that a choice of coordinate system be useful, it usually requires more than being merely a choice of an algebraic basis.
