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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" <>

>>- 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

Stephen J. Herschkorn
Math Tutor on the Internet and in Central New Jersey and Manhattan

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