|
|
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 order-isomorphic?
>>
>>
>
>Not even for totally ordered sets. The integers, the positive
>integers, and the negative integers are three non-isomorphic 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
partial-order 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
|
|
