@PeterBreuer

• An everywhere surjective is a function defined on a topological space where its’ restriction to any non-empty open subset is surjective.
• For natural number n, the n-dimensional interval, for all i∈{1,…n} of real numbers ai and bi is [a1,b1] x [a2,b2] x … x [an,bn]
• An (n+1)-real number is (x1,x2,…,xn+1)∈Rn+1

Peter Breuer I don't know if the post now makes sense (I only studied up to Real Analysis, yet no matter how many times I fail, I continue.)

Suppose:

• For natural number n, set X⊆Rn, set Y⊆R, and (X,P) is a Polish Space
• An everywhere surjective f:X->Y is a function defined on a topological space where its’ restriction to any non-empty open subset is surjective.
• For natural number n, the n-dimensional interval, for all i∈{1,…n} of real numbers ai and bi is [a1,b1] x [a2,b2] x … x [an,bn]

Is it true the graph of an everywhere, surjective f in any n-dim. interval covering the subset of the X x Y, always countably infinite?

If not, is it true the graph of an explicit, bijective f in any n-dim. interval, where the graph of f is dense in the Rn+1, which covers the subset of A x B always countably infinite?

Regarding your comment:

Peter Breuer Oh, well...

Suppose, it does boil down to [what you stated]…(a real valued function of n real arguments…that has the same image set when…restricted to any open subset of the domain of Rn.) Does this mean my first question is correct? If not, is then the second question correct?

Suppose, I say a Polish space is a space homeomorphic to a complete metric space that has a countable dense subset.

I don’t know if that clarified anything. Maybe, someone else can answer.