closure of function

policer

(1) Can sombey give a surjective function change from 3D dimension to 2D that has no "closure"? (2) And visa versa?

Plato

(1) Can sombey give a surjective function change from 3D dimension to 2D that has no "closure"? (2) And visa versa?
closure of function in what sense are you using the word closure here?

policer

Maybe I am wrong but surjective function has a closure definition. Not?

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SlipEternal

Maybe I am wrong but surjective function has a closure definition. Not?
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.

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policer

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?

SlipEternal

1. Yes
2. Yes
3. Read the article I sent you. It will explain what closure is. Once you know what the term means, it should be immediately obvious why it does not have meaning for functions.