Timeline for Does every set $X$ have a topology for which the only continuous self-surjection is the identity map?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 15, 2018 at 4:54 | history | edited | Martin Sleziak | CC BY-SA 3.0 |
added eudml link
|
| Aug 18, 2015 at 13:16 | comment | added | Joel David Hamkins | @AdamPrzeździecki It seems to me that this answer (but using the Vopěnka-Pultr-Hedrlin result, since well-orders wouldn't work without surjectivity) will also answer Dominic's question at: mathoverflow.net/q/215046/1946. Do you agree? | |
| Aug 15, 2015 at 0:32 | comment | added | Eric Wofsey | If the embedding in question does not turn any non-surjection into a surjection (don't know whether this is true, but it seems plausible), then the Vopěnka-Pultr-Hedrlin result is unnecessary, and you can just use ordinals (with their strict ordering). | |
| Aug 14, 2015 at 6:21 | comment | added | Adam Przeździecki | This is about what I got from the paper. I never read it, just skimmed for this answer. She also cites some preprocessing to reduce the problem to connected graphs without loops but I think that this is not necessary. | |
| Aug 14, 2015 at 6:21 | comment | added | Adam Przeździecki | @Joel David Hamkins -- One takes a continuum $C$ constructed by Cook (Eric Wofsey points to it in his comment). It has the properties that any map $C\to C$ is either constant or the identity and for every continuum $H\subseteq C$ other than a point and a nonconstant map $f:H\to C$, $f$ is a retraction onto some $K\subseteq H$. Then you pick two distinct points in $C$ (call them initial and terminal) and replace every edge in $G$ with a copy of $C$. I think that this works fine except it does not prevent edges from collapsing. So she introduces two additional rigid continua... | |
| Aug 13, 2015 at 12:28 | comment | added | Joel David Hamkins | Note also that you use exactly the rigidity result of Vopěnka, Pultr and Hedrlin to which I had alluded in my comment on the original post! | |
| Aug 13, 2015 at 12:27 | comment | added | Joel David Hamkins | Would it be possible for you to sketch the construction? | |
| Aug 13, 2015 at 5:49 | history | answered | Adam Przeździecki | CC BY-SA 3.0 |