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


Re: Open and Shut
Posted:
Feb 3, 2013 4:58 PM


In article <Pine.NEB.4.64.1302022012270.25020@panix3.panix.com>, 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.
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? Zero. ONly {} and Q are both closed and open > Given an closed subset of Q, what's the probablity that it's clopen? Zero. ONly {} and Q are both closed and open 

