How is $\displaystyle \mathbb{R} \cong [0,1] $?
What's the isomorphism?
A little more background would be nice? Do you really mean isomorphism? What kind of isomorphism? Group? Ring? It is surely true that these two sets are equipotent. They are not homeomorphic under the usual topology (this is easy to see since once is compact and the other isn't)? etc.