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My bad. I'll re-read the definitions.The cartesian product is not an n-tuple. The cartesian product is a set of ordered pairs.

ditto"Basis" is a topological term with a specific meaning, and I doubt you mean to use it in that way.

Just about every place I've seen defines the set of ordered pairs that's way, that's why.Why confuse everyone (probably including yourself) when you can use instead? Everyone who recognizes that and are sets will recognize that as their cartesian product.

here's a snippet from a mulivariable calculus book:

I can find many other examples of the Cartesian product defined in just this way. I was merely expressing it the way I've seen it.Suppose A and B are sets. The

Cartesian product AxB of these sets is the collection of all ordered pairs (a,b) such that a is in A and b is in B.

As for the rest, I keep seeing all different kinds of spaces described and defined in various sources. A lot of the terms keep popping up over and over. Why do we need all these different kinds of spaces? Why not just say "space"? There must be some difference or distinction between the different kinds of spaces, and there must be some reason for distinguishing them.

It helps me in attempting to understand what these different kinds of spaces are and how do distinguish between them by comparing and contrasting them, especially because I've read, for example, that a Euclidean space is a metric space and a topological space. So I tried to find an example that was all three and distinguish between the different concepts of space.

As I've said repeatedly, I'm not attempting a formal definition or complete treatment of these topics, only an exercise to help me understand the distinction between them.

Also, I don't appreciate my comments being called "gibberish". This strikes me as pedantic and snobbish, and not in any way instructive or representative of a genuine effort to help me understand.