Does there exist a topological space X with its subspace Y, such that:
-- Y is a Hausdorff subspace;
-- $X \setminus Y$ is a Hausdorff subspace;
-- $\forall y \in Y, \forall x \in X \setminus Y$, there are disjoint X-open subsets $U, V: x \in U, y \in V,$
and X is not Hausdorff itself?
I see that, if Y is open, obviously not. A finite X doesn't exist either. Nevertheless, I suppose that in the general case there is an example. Can anyone suggest any ideas?