# A compact T1 topological space has a proper dense subset to which it is homeomorphic. What can be said about the space?

Let $X$ be a compact T1 (so singleton subsets are closed) topological space. Suppose that there is a proper subset $D \subset X$ such that:

• $D$ is dense in $X$;
• $D$ is homeomorphic to $X$.

Note that $X$ is then not a compactification of $D$, since $D$, being homeomorphic to $X$, is already compact.

My questions:

(1) What can be said about $X$?

(2) Can anything be said about $X$ if either of the compactness or T1 conditions is dropped?

• Could you be more specific? "What can be said" is open-ended. – YCor Jun 15 '15 at 20:01
• This is the very first question I've posted, so please forgive any ambiguity in how I framed the question. I mean, is there a characterization of spaces with this property? My feeling is that perhaps the conditions are too general to say very much at all, but I'm looking for general insight in how to go about answering the question as much as I am looking for a specific answer. – Christian Hoffland Jun 15 '15 at 20:15
• Well, it would then be useful if you say more about what you could think of so far, e.g. trivial remarks and a few examples. – YCor Jun 15 '15 at 20:32
• Obvious comment: $X$ can't be Hausdorff, for then the compact subset $D$ would be closed. – Nate Eldredge Jun 15 '15 at 21:08
• It seems very unlikely to me that much of anything can be said. Note that given any non-compact example, you can get a compact example by adjoining a point whose neighborhoods are the cofinite sets, and I suspect that in many cases you can compactify in other ways as well (in fact, I believe there is always a canonical finest 1-point compactification such that $X\cup\{\infty\}$ is still homeomorphic to $D\cup\{\infty\}$). – Eric Wofsey Jun 16 '15 at 3:58