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