
Re: Partial order and topology
Posted:
Sep 14, 2011 1:17 AM


William Elliot wrote:
> On Tue, 13 Sep 2011, Stephen J. Herschkorn wrote: > >> I have only seen the order topology defined for a totally ordered >> set, but it can be defined for partial orders as well: >> > No, not really, though lots of work has been done on the subject.
? What do you mean by "No"?
> >> Let X be a partially ordered set. Define the order topology to be >> that generated by {{x in X: x < a}, {x in X: x > a}: a in X}. >> > Another topology for a partial order arises with a subbase of the sets > U_a = { x  not x <= a }, V_a = { x  not a <= x } which at least, is T1. > >> Two questions: >>  If X and Y are partially ordered sets that are homeomorphic >> under order topologies, are X and Y necessarily orderisomorphic? > > > No. A linear example, (R,<=) and (R,<=^r) where <=^r = { (x,y)  y <= > x }
Isn't x > x an order isomorphism?
> >>  Is it true that any space can be given a partial order such that >> the topology is the order topology? >> > The topology of a T0 Alexandroff space, a space which any intersection > of open sets is open, can be completely described by the order > a <= b when a in cl {b}. > > Let S = {a,b}. What order topology can you give for (S,indiscrete)?
The empty partial order (using my definition of order topology).
> >  Order and Topology, at.yorku.ca/t/a/i/c/05.htm. > An ordered topological space is a topological space (X,T) equipped with > a partial order <=. Usual compatibility conditions between the topology > and order include convexity (T has a basis of orderconvex sets) and the > T_2ordered property: <=, ie { (x,y)  x <= y }, is closed in XxX. > > which is equivalent to X being Hausdorff. > > Since every topological space (X,T) can be considered as a trivially > ordered space (X,T,=), the theory of ordered spaces includes the usual > topological theory as a special case. Other important cases, each with > its own techniques, include the totally ordered spaces and lattices.
 Stephen J. Herschkorn sjherschko@netscape.net Math Tutor on the Internet and in Central New Jersey and Manhattan

