= Infinite disconnected subspace of a cofinite product
For an infinite <set> $X$ with its <cofinite topology>, every infinite <subset> is <connected>, but $X\times X$ has an infinite disconnected <topological subspace>. Choose $a\ne b$ and use $D=\{a,b\}\times X$. The <open sets> $(X\setminus\{b\})\times X$ and $(X\setminus\{a\})\times X$ have disjoint nonempty traces $\{a\}\times X$ and $\{b\}\times X$ on $D$, covering it. Thus $D$ is a product of a discrete two-point subspace with an infinite cofinite factor. The ambient open rectangles may intersect; only their traces must be disjoint for a <separation> of $D$.
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