Is it true that every compact subset of $\mathbb{R}$ (with a Euclidean metric) is homeomorphic to a closed and bounded interval? Could anyone please give me a hint about how to start to prove that or a counter-example if it's false? It sounds true, since both sets are compact, but it is hard to construct a bijection between two abstract sets.
Thank you in advance.

The set $[0,1]\cup[2,3]$ is compact.
Is it homeomorphic to, say, $[0,1]$?

A classical example which shows that your proposition fails more decisively is the Cantor set.
