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Topic: Open and Shut
Replies: 10   Last Post: Feb 4, 2013 10:50 PM

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William Elliot

Posts: 1,449
Registered: 1/8/12
Re: Open and Shut
Posted: Feb 3, 2013 9:31 PM
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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 non-empty
> order-convex 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 non-empty set of rationals which is (finitely) bounded above
> containing its least upper bound or a non-empty 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?

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