Example of space $T_3$ which is not regular Does there exist example of a topological space in which any closed set and a point can be seperated by open sets i.e. space is $T_3$.
But there exist a pair of points which can't be seperated by points(i.e. points not closed) i.e. space not $T_1$. Hence space not regular because regular space = $T_1 + T_3$

Take the indiscrete topology on any set with more than one point.  Then it's not $T_1$, but it's $T_3$ because any closed set which does not contain a point is empty.
(In fact, any example is basically the same: a $T_3$ space is regular iff it is $T_0$, and a space is $T_3$ iff its $T_0$ quotient is regular.  So the only way to get examples is to take a regular space and add topologically indistinguishable "copies" of points.)
