I have been playing around with the idea of thinking of the set of real valued functions $\mathbb{R}^{\mathbb{R}}$ as the set of uncountably long tuples in $\mathbb{R}$-dimensional space, and I have a question which I have had trouble answering for myself which I would like some help with.
I was wondering how one could construct an explicit bijection from $\mathbb{R}^{\mathbb{R}} \to \mathbb{R}^{(a,b)}$ (for $a,b\in \mathbb{R}$) since $|(a,b)|=|\mathbb{R}|$. My intuition tells me that the bijection should be defined in terms of the range of each function (since that is what distinguishes them), but I have had trouble determining an explicit formula. Also, are there any other interesting bijections that we can obtain, or homeomorphisms if we equip $\mathbb{R}^{\mathbb{R}}$ with the metric $d(f,g)=sup_{x\in \mathbb{R}} \frac{d(fx, gx)}{1+d(fx, gx)}$?
I think it would be interesting to see what we could continuously deform the space of real valued functions (or a subspace of it) into, but I'm having trouble constructing a bijection (let alone a bicontinuous one) to get me started.
Thanks in advance