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

 Messages: [ Previous | Next ]
 Butch Malahide Posts: 894 Registered: 6/29/05
Re: Open and Shut
Posted: Feb 3, 2013 7:52 PM

On Feb 3, 3:58 pm, Virgil <vir...@ligriv.com> wrote:
> In article <Pine.NEB.4.64.1302022012270.25...@panix3.panix.com>,
>  William Elliot <ma...@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.
>
> 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?
>
> 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

Does this mean that Q is connected?

Date Subject Author
2/2/13 William Elliot
2/3/13 Shmuel (Seymour J.) Metz
2/3/13 David C. Ullrich
2/4/13 William Elliot
2/3/13 Virgil
2/3/13 Butch Malahide
2/3/13 William Elliot
2/4/13 Virgil
2/4/13 Butch Malahide
2/4/13 Virgil
2/4/13 Shmuel (Seymour J.) Metz