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

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Virgil

Posts: 6,993
Registered: 1/6/11
Re: Open and Shut
Posted: Feb 3, 2013 4:58 PM
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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 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
--





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