question in real analysis

I'm assuming you're referring to the set of irrational numbers; in that case you can write the irrationals as a union of two disjoint sets, say $\{(\mathbb{R} - \mathbb{Q})\cap (- \infty, 2)\}\cup \{(\mathbb{R} - \mathbb{Q})\cap (2, \infty)\}$. Now recall that a set $A$ is connected iff the only sets that are both open and closed (in $A$) are the empty set and $A$ itself.