Does there exist a non-Hausdorff space with a Hausdorffly located Hausdorff subspace whose extension is also Hausdorff?

Question

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?

Answer 1

Let $X$ be the line with two origins, and let $Y$ be the subset consisting of both of the origins. Note that the only two points of $X$ that cannot be separated by open sets are the two points of $Y$. However, in the subspace topology of $Y$, they can be separated, since each origin has neighborhoods that don't contain the other. Thus $X$ and $Y$ satisfy all of your requirements.