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The task is to prove that $\mathbb{R} \nsim \mathbb{R}^{\mathbb{R}}$.

I know how easy it is to prove this once I can use cardinality, but I can't. I need to show that a bijection between those sets doesn't exist (or by using Cantor-Schröder-Bernstein theorem).

The injection $f:\mathbb{R} \Rightarrow \mathbb{R}^{\mathbb{R}}$ is easy: $$ f(h)=g_h, g_h:\mathbb{R} \Rightarrow \mathbb{R}, g_h(x)=h$$

So, $k(\mathbb{R}) \leq k(\mathbb{R}^{\mathbb{R}})$.

What I struggle to do is to prove that $k(\mathbb{R}) \neq k(\mathbb{R}^{\mathbb{R}})$.

Thanks in advance!

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  • $\begingroup$ Can you show it for $\Bbb R$ vs $2^{\Bbb R}$? $\endgroup$
    – JMoravitz
    Nov 22, 2018 at 21:49
  • $\begingroup$ You mean by showing that $\mathcal{P}(\mathbb{R}) \nsim \mathbb{R}$? $\endgroup$ Nov 22, 2018 at 21:52

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