Re: Stone/Boolean space that is extremally disconnected and is door.

In article <1187654485.109670.81660@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Jose Capco <cliomseerg@xxxxxxxxxxxxxxxxxxxxxxxxx> wrote:

!! On Aug 21, 1:39 am, galathaea <galath...@xxxxxxxxxx> wrote:
!! > In article <1187648304.440703.304...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
!! > Jose Capco <cliomse...@xxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
!! >
!! > !! Dear NG,
!! > !!
!! > !! I am not a topologist, but I do need an example of a space that is:
!! > !!
!! > !! Boolean (or Stone), Extremally disconnected and has only one
!! > !! nonisolated point.
!! > !!
!! > !! I have looked at papers that may give me possible answers to this, but
!! > !! I cannot conclude at once that their examples have the property I am
!! > !! searching. I am not sure now if this question is actually "trivial",
!! > !! but I don't suppose it is :) .. would appreciate any help.
!! >
!! > any discrete space is both door and extremally disconnected
!!
!! but not Boolean! Because not any discrete space is compact.
!!
!! Boolean is equivalent to : Compact, Hausdorff and Totally
!! disconnected.

but finite discrete spaces are compact

i know it seems a trivial example
but i not only think it satisfies your criteria
i think it has been used for these properties in work on denotational semantics

in computational domain theory
properties like kuratowski-finiteness and extremally-disconnectedness
become important for establishing fixed point theorems
and i am sure i have seen work that starts with finite discrete boolean spaces
and lifts these properties to more exotic spaces

i am sure i have seen them explicitly use that these are extremally disconnected and door
but i could be mistaken...?

there is one book i want to check when i get home...

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galathaea: prankster, fablist, magician, liar
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