# 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 = $T_1 + T_3$

• @bof no these not defined like this. I was reading Kelley book. I write like these to specify what I need t ask. – Sushil Sep 20 '16 at 8:32

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.)
• I would have said it's regular but not T$_3.$ It seems that sometime after I graduated some vandal went around and changed all the definitions. – bof Sep 20 '16 at 8:33
• @bof: Both conventions are in use, but I’d like to have words with the idiot who came up with the OP’s: the $T_k$ notation was obviously intended to be a hierarchy. – Brian M. Scott Sep 20 '16 at 19:56