Example of space that is separable but not a Hausdorff space. 
Give an example of a topological space that is separable but not a Hausdorff space. 

I have not been able to discover an example, I thought of the Arens-Fort space because this is separable, since this space is countable and the same set is a countable dense about yourself, but do not know how to show that is second-countable. This space serves as an example, there are some other more. Thanks
 A: Just take a finite space that isn't Hausdorff, such as the set $X=\{s,\eta\}$ with the topology where the open sets are $X$, $\emptyset$, and $\{\eta\}$. 
A: $\pi$-Base, which draws its data from Steen and Seebach's Counterexamples in Topology, lists the following separable, non-Hausdorff spaces. You can learn more about these spaces by viewing the search result.
Compact Complement Topology
Countable Excluded Point Topology
Countable Particular Point Topology
Deleted Integer Topology
Divisor Topology
Double Pointed Reals
Finite Complement Topology on a Countable Space
Finite Complement Topology on an Uncountable Space
Finite Excluded Point Topology
Finite Particular Point Topology
Hjalmar Ekdal Topology
Indiscrete Topology
Interlocking Interval Topology
Maximal Compact Topology
Nested Interval Topology
Odd-Even Topology
One Point Compactification fo the Rationals
Overlapping Interval Topology
Prime Ideal Topology
Right Order Topology on the Reals
Sierpinski Space
Telophase Topology
The Integer Broom
Uncountable Particular Point Topology
A: There are many examples. The simplest examples are as following:
1, Indiscrete Topology
2, Finite Complement Topology on a countably infinite set
A: Since any set X is countable if there exists a bijective map from X into subset of N 
If m is fixed +ve integer such that (12345..........m) is that subset of N then m is the cardinality of X,nd X is countable.
So here i forced to say that the finite complement topology on an countable (infinite) set is not Hausdorff .
If X is finite then (X,Tf) is discrete topology so becomes Hausdorff.
