Let $F$ be the set of all irrational numbers in $\mathbb {R}$. $X=\mathbb {R^2}(F×F )$ with the usual subspace topology of $\mathbb {R^2}$.Number of connected components of $X$?
I know only that if $A$ is countable then $\mathbb{R^n}A$ is connected. But how that will help me . Please help.
Write $X = \mathbb{Q} \times \mathbb{R} \cup \mathbb{R} \times \mathbb{Q}$. Since $X$ is covered by lines in both coordinate directions, it is path connected. In particular $X$ has one connected component.