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

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Virgil

Posts: 8,833
Registered: 1/6/11
Re: Open and Shut
Posted: Feb 4, 2013 12:34 AM
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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 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.


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 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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