I was asked the following:
Are there any infinite subsets of $X \times X$ which are not connected? $X$ is given the cofinite topology, and $X \times X$ is given the product topology.
edit: $X$ is connected, as there are no disjoint nonempty open connected sets. But this holds too for the basis elements of $X \times X$ isn't it? So all such subsets are connected?