Timeline for On the classification of second-countable Stone spaces
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13 events
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| Feb 15 at 19:43 | comment | added | მამუკა ჯიბლაძე | @SimonHenry I see, sorry. I had in mind my previous comment about atomic countable Boolean algebras. Representing them as a quotient of a free Boolean algebra dually they correspond to a closed subset of the Cantor space; but since they are generated by atoms dually you get scattered closed subsets, i. e. those with the subset of isolated points dense. | |
| Feb 15 at 18:21 | comment | added | Simon Henry | @მამუკაჯიბლაძე I'm not sure I understand what you are saying. By Stone duality, Given a boolean algebra B, closed subspaces of $Spec(B)$ are in correspondence with quotient of $B$ (as a boolean algebra). So you get all closed subspace this way. (to be clear I wasn't answering your comment, I was talking about Tim's answer) | |
| Feb 15 at 3:48 | comment | added | მამუკა ჯიბლაძე | @SimonHenry Only very special closed subspaces, and I am failing to characterize them. For example, they cannot have perfect core, i. e. iterating their derived subsets transfinitely should eventually reach empty set. In fact maybe their Cantor-Bendixson rank does not exceed $\omega$? | |
| Feb 12 at 18:59 | comment | added | Simon Henry | That is interesting, but one should note that given that the dual of any countable boolean algebra can be realized as a closed subset of the Cantor space, I'm not completely sure this classifies anything: Classyfing Filtration of the cantor space by closed subspaces up to homomorphisms subsumes classifying countable boolean algebra. | |
| Feb 12 at 16:49 | comment | added | მამუკა ჯიბლაძე | Ouch. Seems these are very special continuous images. What I was trying to use is this. Let $B$ be an atomic BA, then assigning to an element of $B$ the set of atoms below it is an embedding of $B$ into the powerset of atoms. So if $B$ is also countable it embeds into the powerset of a countable set. Dually we get a continuous onto from the dual of the powerset to the dual of $B$. And dual of the powerset is the Stone-Čech. It seems I am overlooking something very special about that embedding, or dually about that onto map... | |
| Feb 12 at 15:08 | comment | added | Tim Campion | @მამუკაჯიბლაძე The Stone Cech compactification of an infinite set is never second countable. | |
| Feb 12 at 12:33 | comment | added | მამუკა ჯიბლაძე | Might be useful to add concerning the scattered part that duals of scattered spaces are precisely atomic Boolean algebras (every nonzero element has an atom below it). I believe this also gives that second countable scattered Stone spaces are precisely continuous images of the Stone-Čech compactification of a countable discrete space. But maybe this is directly obvious too... | |
| Feb 12 at 5:36 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
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| Feb 12 at 4:07 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Feb 12 at 3:53 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Feb 12 at 2:09 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Feb 12 at 2:08 | history | made wiki | Post Made Community Wiki by Tim Campion | ||
| Feb 12 at 1:57 | history | answered | Tim Campion | CC BY-SA 4.0 |