Discrete Space is Extremally Disconnected
Then $T$ is extremally disconnected.
First we note that as Discrete Space satisfies all Separation Properties, $T$ is a $T_2$ (Hausdorff) space.
Then from Interior Equals Closure of Subset of Discrete Space, it follows directly that the closure of every open set of $T$ is open.
Hence by definition $T$ is extremally disconnected.