Is $\mathbb{R} \setminus \mathbb{Q}$ connected?

Question

In case of $\mathbb{R} \setminus \mathbb{Q}$ we can construct two open sets union of which is $\mathbb{R} \setminus \mathbb{Q}$ but here we eliminate infinitely many uncountable points then how we can construct open sets? I am confused. Sorry for language error. Thank you...

Answer 1

$\mathbb{R}$ will become disconnected if you remove even a single point. For example, $\mathbb{R} \setminus \{3\}$ is disconnected because you can write it as a union $A \cup B$ where $A = (-\infty,3)$ and $B = (3, \infty)$ are two disjoint open sets.

The only way to remove stuff from $\mathbb{R}$ and get a connected set is if what's left is an interval. Otherwise, there'll always be two remaining points $a < b$ with a removed point $c$ in between, and you can use the same setup as above to break your set $S$ into nonempty open subsets $S \cap (-\infty,c)$ and $S \cap (c,\infty)$.

In general, the idea of "connected" is that it shouldn't be possible to break the set into two pieces that don't touch each other.

(Thanks @DanishChef for pointing out a math error in an earlier version of this answer.)