Pick an \(n\in\{0,1,2,3\}\), and give an example of a space that is \(T_n\), but not \(T_{n+1}\).
Solution
Take \(n=0\). Let \[ X=\{0,1\},\qquad \tau=\{\varnothing,\{1\},X\}, \] the Sierpiński topology.
The space is \(T_0\): the open set \(\{1\}\) contains \(1\) but not \(0\), so the two points are topologically distinguishable.
It is not \(T_1\): the singleton \(\{1\}\) is not closed, since its complement \(\{0\}\) is not open. Equivalently, there is no open neighborhood of \(0\) that omits \(1\), because the only open neighborhood of \(0\) is \(X\).
Hence this is a \(T_0\) space that is not \(T_1\).