
Re: Partial order and topology
Posted:
Sep 15, 2011 3:40 PM



Butch Malahide wrote:
>On Sep 13, 10:45 pm, "Stephen J. Herschkorn" <sjhersc...@netscape.net> >wrote: > > >> If X and Y are partially ordered sets that are homeomorphic under >>order topologies, are X and Y necessarily orderisomorphic? >> >> > >Not even for totally ordered sets. The integers, the positive >integers, and the negative integers are three nonisomorphic ordered >sets, but they all have the discrete topology. > > > >> Is it true that any space can be given a partial order such that the >>topology is the order topology? >> >> > >There are 4 distinct topologies (3 nonhomeomorphic ones) on the set >{0,1}, but only 3 distinct partial orders (2 nonisomorphic ones). So I >would say the answer is no. > >
So I alter the question: Can any topological space be embedded in a partialorder space? For example, ({0, 1}, {{}, {0}, {0, 1}}) can be embedded in the space {0, 1, 2} with the partial order {(0, 2)}.
(Remember that a subspace of a topological space with the order topology does not necssarily have the topology from the order restricted to that subset.)
 Stephen J. Herschkorn sjherschko@netscape.net Math Tutor on the Internet and in Central New Jersey and Manhattan

