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It will have to be non-$T_1$. The Sierpinski space with the closed point as $Y$ would be the minimal example I think, if not very interesting.
There are probably a lot of examples with finite topological spaces, I donâ€™t know if they are interesting though. Oh and every space should be codense in itself?
So no intuitive story such as in the dense case of how behaviour on $\mathbb{Q}$ determines behaviour on $\mathbb{R}$?
A quick search came up with that the axiom of choice is equivalent to every non-empty space having a codense T_0 subspace. The name of McCarten was mentioned.
There is a related notion in locale theory see http://www.mat.uc.pt/preprints/ps/p1445.pdf
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