Virgil
Posts:
4,479
Registered:
1/6/11
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Re: Open and Shut
Posted:
Feb 4, 2013 3:08 AM
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In article
<aef313bf-935a-47e5-a5f9-1781ba694ce8@h17g2000yqe.googlegroups.com>,
Butch Malahide <fred.galvin@gmail.com> wrote:
> 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:
> >
> >
> >
> >
> >
> > > 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.
Oops! My error! I stand corrected!
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