# Thread: closure of function

1. ## closure of function

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

2. ## Re: closure of function

Originally Posted by policer
(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?

3. ## Re: closure of function

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

4. ## Re: closure of function

Originally Posted by policer
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.

5. ## Re: closure of function

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?

6. ## Re: closure of function

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.