Timeline for Characterization of Stone-Cech compactifications
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 23, 2012 at 19:31 | vote | accept | Noah Schweber | ||
| Sep 23, 2012 at 14:58 | comment | added | Todd Trimble | Ah, I think I see. So the compact Hausdorff extremal disconnectedness of $X$ means the clopens of $X$ form a complete Boolean algebra $B$. The open points are atomic elements in $B$, and the density should imply that every clopen is a join in $B$ of such atomic elements. $B$ is therefore a complete atomic Boolean algebra, and thus of the form $PS$ for some set $S$. The Stone space of $B$ (which is $X$) is thus the Stone space of $PS$, which is $\beta(S)$ essentially by definition. Thanks! | |
| Sep 23, 2012 at 13:02 | comment | added | Simon Henry | actualy my idea come more from the fact that compact, hausdorf, extremally disconected space (stonean space) are exactly the stone-Cech compactification of boolean local (I think it's in Johnstone's Stone Space too, isn't it ? ). So i just had to add an hypothesis of existence of "points" for the local of clopen to be able to conclude that the starting boolean local was a discret space. But i guess those two result are closely related. | |
| Sep 23, 2012 at 12:10 | comment | added | Todd Trimble | +1, this is interesting. Andrew Gleason proved that the projective objects in compact Hausdorff spaces (i.e., retracts of free objects, i.e., retracts of Stone-Cech compactifications of discrete spaces) are exactly the compact, Hausdorff, extremally disconnected spaces (this result can be found in Johnstone's Stone Spaces). | |
| Sep 23, 2012 at 7:54 | history | answered | Simon Henry | CC BY-SA 3.0 |