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Let $X$ be an arbitrary set. I consider the set of all real functions defined on $X$. I know that this is usually denoted by $\mathbb{R}^X$. However, I am interested in characterizing each point of $\mathbb{R}^X$, that is I would like to write it in the form $$ \mathbb{R}^X=\{x\mid \dots \}.$$

How could one do that?

Rachel
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  • $x$ is a function, and its domain is $X$, and for each $t\in X$, we have that $x(t)\in\mathbb R$. Maybe saying all this looks awkward; one usually codes all of it as $x:X\to\mathbb R$. If you feel ${x\mid x:X\to\mathbb R}$ is awkward as well, a compromise such as ${x\mid x$ is a function, $x:X\to\mathbb R}$ should work. – Andrés E. Caicedo Nov 30 '13 at 18:57

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If you really want to do such a thing, you can write

$${}^X\Bbb R=\{x\subseteq X\times\Bbb R:\forall \xi\in X\,\exists!r\in\Bbb R(\langle\xi,r\rangle\in x)\}\;.$$

(I prefer ${}^XY$ for the set of functions from $X$ to $Y$.) In case you’ve not seen the quantifier $\exists!$ before, it means there is exactly one. The condition $\forall\xi\in X\,\exists!r\in\Bbb R(\langle\xi,r\rangle\in x)$ can be expanded to

$$\forall\xi\in X\,\exists r\in\Bbb R\Big(\langle\xi,r\rangle\in x\land \forall s\in\Bbb R(\langle\xi,s\rangle\in x\to s=r)\Big)\;.$$

Brian M. Scott
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  • Thank you. My teacher wrote $$\mathbb{R}^X={x=x(t):t\in X, x(t) \in \mathbb{R}}.$$ Is this in any way correct? Isn't it a little sloppy? – Rachel Dec 01 '13 at 00:56
  • @Rachel: I consider it very sloppy indeed and would not write such a thing. At that level of informality one might better simply say that ${}^X\Bbb R$ or $\Bbb R^X$ is the set of functions from $X$ to $\Bbb R$; the (bad) notation adds nothing to that, either in comprehensibility or in precision. – Brian M. Scott Dec 01 '13 at 01:13