Must a space homeomorphic to $\Bbb R\setminus \Bbb Q$ have a countable complement?
There is a problem from a list suggested practice problems that I am having issues with. It says:
Suppose that $X$ is a subspace of the real line $\mathbb{R}$ which is homeomorphic to the space of irrational numbers. Is the complement of $X$ in $\mathbb{R}$ necessarily countable?
Would anyone be willing to help me out with this one? Thank you so much!
Hint: $\mathbb R$ is homeomorphic to $(0,1)$.