We endow $\mathbb{R}$ by the natural Topology
Let $E$ be a an infite set of $\mathbb{R}$,such that $E \subset [0, 1]$.
Do we have this property :
As the cardinal of $E$ is infinite, the closure of $E$ denoted $cl(E)$ is an interval $I$ such that $ cl(E)=I \subseteq [0, 1]$ ?
Any suggestions or references are welcome
The Cantor set is a compact subset of $[0,1]$ that contains no interval but is uncountable. So no.