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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
Partial order and topology
Posted: Sep 13, 2011 11:45 PM
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I have only seen the order topology defined for a totally ordered set,
but it can be defined for partial orders as well:

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}.

Two questions:
- If X and Y are partially ordered sets that are homeomorphic under
order topologies, are X and Y necessarily order-isomorphic?
- Is it true that any space can be given a partial order such that the
topology is the order topology?

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

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