??so what about double negation topos of localic topos corresponding to stone space?? ... ??for example cantor space ... ?? and so forth ... ???...

??relevance to conjecture about "t-model with every t-model morphism out of it an elementary equivalence" .... ???...

confusion ... ??...

regularly open sets in a top space form.... ???what??? ... ???a complete boolean algebra??? ... ??or what ???...


consider for example 1-point compactification of natural numbers...

??regular open =?= ... ??? finite not containing infinity, or complement thereof?? ...

(hmm, i guess that below we refute this... or something ... ???so some of the confusion here had to do with... ??not noticing that ... ???there's a whole lot of infinite but not cofinite clopen sets here??? ???or something ??? no, wait, that still seems wrong... still lots of confusion here ....??? ... ??what about whether clopen is equivalent to regular open here ?????..... ??something about ... if a subset is neither finite nor cofinite, then it's open xor closed... depending precisely on whether it contains infinity ... ???or something ???.... ??and thus not clopen??? )


supremum of the finite regular opens = ... ???non-existent ????

(note added later: not sure that makes any sense even on its own mistaken terms ... ???or something ???... )

hmmm.... ???....

?????.....

??closed = finite or contains infinity??

??open = cofinite or lacks infinity??

??every closed set x here is the closure of an open set ??? ???namely of "x minus infinity" ??.... ??no, wait... a finite set containing infinity is not the closure of x minus infinity ... in fact is not the closure of any open set... neither of a cofinite set (pretty obviously...) nor of a set lacking infinity (almost as obviously...)....

??so "closure of open" here = "finite xor containing infinity" ??? ??...

??so then is it true that the double negation topos here is the localic topos correponding to ... ???the (discrete) subspace obtained by removing infinity ??? or something ????.... hmmmm.... ????what about something about "perfect" (or something) here ???? .....


??????what about boolean algebra of "finite or cofinite subsets of the natural numbers" ???.....

???so does 1-point compactification of natural numbers have lots of regular opens that aren't clopen???

for example, consider the evens not including infinity ... the closure is the evens including infinity... the interior of that is ... ??the evens not including infinity ... so it seems regular open ... but certainly not clopen, right ???... because not closed ... ???

???at the moment seems really stupid to have imagined that for stone spaces clopen might be equivalent to regular open ... ???did we really imagine it then?? ???perhaps yes?? .... ??and could this explain some bad recurring confusion that we had ??? or something ????.....


??so maybe ... ???the clopens in a stone space form "the" boolean algebra .... ???not necessarily complete .... ??and the regular opens form ... ???a complete boolean algebra .... ???which is the global points of the subobject classifier in the double negation topos of the localic topos corresponding to the stone space ??? ????or something ??? is that right ???....

??regular opens as maybe obviously co-/complete by gabriel-ulmer stuff (or something... and so forth ... ???) ??something about ... any lawvere-tierney topology .... ???....

??in e-mail to toby from around january 2010 i seem to be suggesting stuff like "in a stone locale (??...), regular open implies clopen" or something ... ??and also talking about something called "stone algebra" and / or "stone lattice" ....