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Re: Open and Shut
Posted:
Feb 4, 2013 1:22 AM
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On Feb 3, 11:34 pm, Virgil <vir...@ligriv.com> wrote:
> In article <Pine.NEB.4.64.1302031827150.2...@panix1.panix.com>,
> William Elliot <ma...@panix.com> wrote:
>
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>
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> > On Sun, 3 Feb 2013, Virgil wrote:
> > > 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.
>
> > (-pi,pi) /\ Q is a proper, not empty, clopen, order convex subset of Q.
>
> Depends on which topology one has for Q.
There is only one "natural" topology for Q. The order topology of Q
coincides with the subspace topology of Q as a subspace of R. This is
not true for every subset of R (e.g. consider [0,1) union [2,3)); it
is true for Q because Q is a dense subset of R.
> 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.
Wrong. It is true that the open intervals of Q with rational endpoints
constitute a base for the topology of Q (a vector space has a BASIS, a
topology has a BASE), and the set (-pi, pi) /\ Q is not an element of
that base, but it is an open set in Q because it is a union of
elements of the base. This is analogous to the fact that, in the more
familiar setting of the real line, the collection of all intervals
with rational endpoints is a base for the topology, while an open
interval with one or both endpoints irrational is also an open set,
being a union of open intervals with rational endpoints.
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