Included-point and finite-complement topologies on $\mathbb R$ are not homeomorphic

Let \(X=Y=\mathbb R\). Give \(X\) the included-point topology with included point \(0\), and give \(Y\) the finite-complement topology. Determine whether \((X,\mathcal T_{ip})\) and \((Y,\mathcal T_{fc})\) are homeomorphic, and prove your answer.