(1) Can sombey give a surjective function change from 3D dimension to 2D that has no "closure"? (2) And visa versa?
https://en.wikipedia.org/wiki/Closure_(mathematics)
Please read what the word Closure means. There is not a single reference in the article about functions, surjective or otherwise.
Perhaps you mean a surjective function that is nowhere continuous? And a surjection from 2D to 3D that is nowhere continuous?
Let $C:\mathbb{R} \to \mathbb{R}$ be the Conway Base-13 function.
Then define $f:\mathbb{R}^3 \to \mathbb{R}^2$ by $f(x,y,z) = (C(x),C(y))$. This is a surjection that is nowhere continuous.
Define $g:\mathbb{R}^2 \to \mathbb{R}^3$ by $g(x,y) = (C(x),C(y),C(C(x)))$. This is probably a surjection that is nowhere continuous.
But surjective function has a limitted definition. The term closure not except here.
If I go from 3D to 2D, I think I loss information.
My questions - (1) if it is true?
(2)If I go from 2D to 3D, I need to give more information. Right?
(3) Is it really, that the word closure is not used in function definition. Why?