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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 = $T_1 + T_3$

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

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  • $\begingroup$ 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. $\endgroup$ – bof Sep 20 '16 at 8:33
  • $\begingroup$ I'm just following OP's usage; I've seen both conventions and tend to be apathetic about which to use (I very rarely want to refer the version that doesn't include points being closed anyways). $\endgroup$ – Eric Wofsey Sep 20 '16 at 8:35
  • $\begingroup$ @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. $\endgroup$ – Brian M. Scott Sep 20 '16 at 19:56

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