On Thursday, January 23, 2020 at 8:59:18 AM UTC-7, Lawrence Crowell wrote: > > On Thursday, January 23, 2020 at 8:47:25 AM UTC-6, John Clark wrote: >> >> On Thu, Jan 23, 2020 at 6:40 AM Alan Grayson <[email protected]> wrote: >> >> >>Lawrence Crowell wrote: *I would say the spatial surface is >>>> topologically closed, but not causally closed. * >>> >>> >>> *> As I just posted, this is correct, but can you give a precise >>> mathematical meaning to "topologically closed"? TIA, AG * >>> >> >> The Universe is topologically closed if you can give me any point in the >> universe I can give you a number greater than zero that you can use as a >> radius to draw a sphere centered on that point such that every point within >> that sphere is also in the universe. Or to put it more succinctly, if the >> universe is topologically closed then it would contain all its limit >> points. But you want to know if it's finite or infinite and this would tell >> you nothing about that. >> >> John K Clark >> > > AKA Heine-Borel theorem. >
But how does this definition of "topologically closed" distinguish a sphere from a plane? Nor does connectedness help. AG > > LC > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/a59a44f6-f4b9-462a-9134-7c28cbacf3f0%40googlegroups.com.
