Show that if \(X\) is an infinite set, it is connected in the finite complement topology.
Solution
In the finite-complement topology on an infinite set \(X\), every nonempty open set has finite complement.
Suppose \(U,V\) were disjoint nonempty open sets. Then \[ U\subseteq X-V, \] so \(U\) is finite because \(X-V\) is finite. But \(U\) is nonempty open, so \(X-U\) is also finite. Therefore \[ X=U\cup(X-U) \] would be finite, a contradiction. Thus no separation exists, and \(X\) is connected.