Q:

Why are $\mathbb{Q}$ and $\mathbb{R}\setminus \mathbb{Q}$ not homeomorphic to $\mathbb{R}$?

Why are $\mathbb{Q}$ and $\mathbb{R}\setminus \mathbb{Q}$ not homeomorphic to $\mathbb{R}$?

Does it have something to do with the open sets in $\mathbb{Q}$ and $\mathbb{R}\setminus \mathbb{Q}$ or the density? Or even the cardinality?

A:

Since $\mathbb Q$ and $\mathbb R$ do not have the same cardinal, they cannot possibly be homeomorphic.
And $\mathbb R\setminus\mathbb Q$ is not connected. Therefore, it is also not homeomorphic to $\mathbb R$.

