Topological space example

Question

What would be an example of topological space having any two points say $x_1, x_2$, such that each open set containing one point ( $x_1$) also contains the other point $x_2$

Answer 1

Look at the trivial topology $\{ \emptyset, X \}$ for any set $X$ that has at least two members.

Answer 2

The property you describe is related to the Hausdorff property.

Definition: A topological space $X$ is Hausdorff if for any $x,y\in x$ with $x\neq y$ there are open neighbourhoods $U_x$, $U_y$ of $x$ resp $y$ such that $U_x \cap U_y = \emptyset$

So every space fulfilling your conditions hast to be non-Hausdorff.

There are many examples of non-Hausdorff spaces, but they are all a bit unintuitive as we can't separate points by open sets.
Famous examples are the trivial topology or the Zarisky topology.

You can check if these topologies fulfil your requirements.