1

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?

Marios Gretsas
  • 21,906

1 Answers1

4

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 $\mathbb R$.

José Carlos Santos
  • 427,504
  • Another approach: Cauchy sequences always converge in $\mathbb R$ (but not so in the other two spaces). – hardmath Oct 04 '19 at 11:20
  • 7
    @hardmath This is a wrong argument. Cauchy sequences converge in $\mathbb{R}$ but not in $(0,1)$, though both spaces are homemorphic. Metric properties are not preserved by homeomorphisms.... – J. De Ro Oct 04 '19 at 11:41
  • @EpsilonDelta: Your observation is true, but a correct argument can be based on local compactness (as José Carlos Santos pointed out in a now removed Comment) or on the topological property of "being homeomorphic to a complete metric space". – hardmath Oct 04 '19 at 19:48
  • 1
    @hardmath Also, the irrationals are neither locally compact nor $\sigma$-compact. They're very different from the reals. – Henno Brandsma Oct 04 '19 at 22:14