Virgil
Posts:
8,833
Registered:
1/6/11


Re: Open and Shut
Posted:
Feb 4, 2013 12:34 AM


In article <Pine.NEB.4.64.1302031827150.2987@panix1.panix.com>, William Elliot <marsh@panix.com> wrote:
> 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.
Depends on which topology one has for Q.
If one uses the order topology on Q, in which a basis of the interiors of intervals with ENDPOINTS IN Q, Not in R, then your set is not closed in 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? > >  

