9
$\begingroup$

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?

Improve this question
$\endgroup$
9
  • 4
    $\begingroup$ Could you be more specific? "What can be said" is open-ended. $\endgroup$
    – YCor
    Commented Jun 15, 2015 at 20:01
  • $\begingroup$ 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. $\endgroup$
    – Christian Hoffland
    Commented Jun 15, 2015 at 20:15
  • $\begingroup$ 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. $\endgroup$
    – YCor
    Commented Jun 15, 2015 at 20:32
  • 2
    $\begingroup$ Obvious comment: $X$ can't be Hausdorff, for then the compact subset $D$ would be closed. $\endgroup$
    – Nate Eldredge
    Commented Jun 15, 2015 at 21:08
  • 2
    $\begingroup$ 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 $D\cup\{\infty\}$). $\endgroup$
    – Eric Wofsey
    Commented Jun 16, 2015 at 3:58

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .