Show that this is not T_{1} -space.

Let X be the three-element set {a,b,c}, and let the topology consist of the following subsets of X:empty set,{a},{a,b},{a,c},X.
Show that this is not T_{1} -space.

Note that a and c are not separable; that is, there do not exist open sets in X which does not contain the other point (since any of the open sets in X containing c also contains a).

Hence, X is not T_1.

I hope this helps!