partial ordering in a topological space
Proposition 1.
is a preorder.
Proof.
Clearly . Next, suppose and . Let be a closed set containing . Since is in the closure of , . Since is in the closure of , also. So . ∎
We call the specialization preorder on . If , then is called a specialization point of , and a generization point of . For any set ,
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the set of all specialization points of points of is called the specialization of , and is denoted by ;
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the set of all generization points of points of is called the generization of , and is denoted by .
Proposition 2.
. If is (http://planetmath.org/T0), then is a partial order.
Proof.
Suppose next that and . If , then there is an open set such that and . So , the complement of , which is a closed set. But then since it is in the closure of . So , a contradition. Thus . ∎
This turns a topological space into a poset, where here is called the specialization order of the space.
Given a space, we have the following:
Proposition 3.
iff implies for any open set in .
Proof.
if and , then . Since , we have , a contradiction. if , then for some closed set , we have and . But then , so that , a contradiction. ∎
Remarks.
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, the lower set of . ( iff iff ).
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But if is (http://planetmath.org/T1), then the partial ordering just defined is trivial (the diagonal set), since every point is a closed point (for verification, just modify the antisymmetry portion of the above proof).
Title | partial ordering in a topological space |
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Canonical name | PartialOrderingInATopologicalSpace |
Date of creation | 2013-03-22 16:35:02 |
Last modified on | 2013-03-22 16:35:02 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 10 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 54F99 |
Defines | specialization order |
Defines | specialization preorder |
Defines | specialization |
Defines | generization |