Virgil
Posts:
9,012
Registered:
1/6/11


Re: Open and Shut
Posted:
Feb 4, 2013 3:08 AM


In article <aef313bf935a47e5a5f91781ba694ce8@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 nonempty > > > > orderconvex 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! 

