
Re: Open and Shut
Posted:
Feb 3, 2013 9:31 PM


On Sun, 3 Feb 2013, Virgil wrote: > William Elliot <marsh@panix.com> wrote: > > > A subset A, of an ordered set is convex when > > for all x,y in A, for all a, (x <= a <= y implies a in A). > > > > I will call an interval an order convex subset of Q. > > Given an interval, what's the probablity that it's > > open, closed, both, neither? > > The only probability that is certain in Q is that the probability of > being both open and closed is zero, as Q and {} are the only nonempty > orderconvex sets in Q that are both open and closed under the order > toology, and there are infinitely many other intervals which are not > both open and closed. (pi,pi) /\ Q is a proper, not empty, clopen, order convex subset of Q.
> To do more one needs to make some assumptions about the probability > of a nonempty set of rationals which is (finitely) bounded above > containing its least upper bound or a nonempty set of rationals which > is (finitely) bounded below containing its greatest lower bound.
> > Given an open subset of Q, what's the probablity that it's clopen?
> > Given an closed subset of Q, what's the probablity that it's clopen?


