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Topic: Partial order and topology
Replies: 7   Last Post: Sep 16, 2011 6:25 PM

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Stephen J. Herschkorn

Posts: 2,297
Registered: 1/29/05
Re: Partial order and topology
Posted: Sep 14, 2011 1:17 AM
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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 order-isomorphic?

> 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,
> 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 order-convex sets) and the
> T_2-ordered 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
Math Tutor on the Internet and in Central New Jersey and Manhattan

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