Alexandroff space is T1 if and only if it is discrete
Proposition. Let be an Alexandroff space. Then is if and only if is discrete.
Proof. ,,” It is easy to see, that every discrete space is Alexandroff and .
,,” Recall that topological space is if and only if every subset is equal to the intersection
of all its open neighbourhoods. So let . Then the intersection of all open neighbourhoods of is equal to . But since is Alexandroff, then is open and thus points are open. Therefore is discrete.