A T1 Space That is Not a T2 Space
Consider the set
Define the topologyon
Sets containing 1 are open if and only if they are the complement of a finite set.
Sets not containing 1 are open when they are open in setr with the usual topology.
Hence, each open set containing 0 is infinite.
Therefore points 0 and 1 cannot be placed in open disjoint sets.
The space is T1 but not T2.