Topology generated by subsets with infinite complement

For a set \(X\), define \[ \mathcal B:=\{U\subseteq X:X\setminus U\text{ is infinite}\}\cup\{X\}. \]

1. Show that \(\mathcal B\) is a basis for a topology on \(X\).

2. Identify the topology \(\mathcal T(\mathcal B)\) for every set \(X\), and prove your answer.