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?