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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 15, 2011 3:40 PM
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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





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