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• CommentRowNumber1.
• CommentAuthorDavid_Corfield
• CommentTimeJan 29th 2019

A stub here. Is there a ready example?

• CommentRowNumber2.
• CommentAuthorRichard Williamson
• CommentTimeJan 29th 2019
• (edited Jan 29th 2019)

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.

• CommentRowNumber3.
• CommentAuthorAlizter
• CommentTimeJan 29th 2019
• (edited Jan 29th 2019)

There are probably a lot of examples with finite topological spaces, I don’t know if they are interesting though. Oh and every space should be codense in itself?

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeJan 29th 2019

So no intuitive story such as in the dense case of how behaviour on $\mathbb{Q}$ determines behaviour on $\mathbb{R}$?

• CommentRowNumber5.
• CommentAuthorTim_Porter
• CommentTimeJan 29th 2019
• (edited Jan 29th 2019)

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