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partial ordering in a topological space

(Definition)

Let be a T0 space. For any $x,y\in X$ , we define a binary relation $\le$ on as follows:

$\displaystyle x\le y$ iff $\displaystyle x \in \overline{\lbrace y\rbrace}.$

Proposition. The binary relation just defined is a partial order.

Proof. Clearly $x\le x$ . Suppose next that $x\le y$ and $y\le x$ . If $x\ne y$ , then there is an open set

such that $x\in A$ and $y\notin A$ . So $y\in A^c$ , the complement of

, which is a closed set. But then $x\in A^c$ since it is in the closure of $\lbrace y\rbrace$ . So $x\in A\cap A^c=\varnothing$ , a contradition. Thus

. Finally, suppose $x\le y$ and $y\le z$ . Let

be a closed set containing

. Since

is in the closure of $\lbrace z\rbrace$ , $y\in C$ . Since

is in the closure of $\lbrace y\rbrace$ , $x\in C$ also. So $x\le z$ . $\qedsymbol$

This turns the topological space into a poset.

$\le$ is called the specialization order of . We have the following

$x\le y$ iff $x\in U$ implies $y\in U$ for any open set in

Proof. $(\Rightarrow):$ if $x\in U$ and $y\notin U$ , then $y\in U^c$ . Since $x\le y$ , we have $x\in U^c$ , a contradiction. $(\Leftarrow) :$ if $x\notin \overline{\lbrace y\rbrace}$ , then for some closed set

, we have $y\in C$ and $x\notin C$ . But then $x\in C^c$ , so that $y\in C^c$ , a contradiction. $\qedsymbol$

Remarks.

If is any topological space, then $\le$ is merely a preorder.
$\overline{\lbrace x\rbrace}=\downarrow x$ , the lower set of . ( $z\in \downarrow x$ iff $z\le x$ iff $z\in\overline{\lbrace x\rbrace}$ ).
But if is a Hausdorff space, then the partial ordering just defined is trivial (the diagonal set), since singletons in are closed (for verification, just modify the antisymmetry portion of the above proof).

"partial ordering in a topological space" is owned by CWoo.

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See Also: specialization

Also defines:

specialization order

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Cross-references: proof, antisymmetry, closed, singletons, diagonal, Hausdorff space, lower set, preorder, contradiction, implies, iff, poset, topological space, closure, closed set, complement, open set, partial order, proposition, binary relation, T0 space
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This is version 5 of partial ordering in a topological space, born on 2007-01-16, modified 2007-01-18.
Object id is 8775, canonical name is PartialOrderingInATopologicalSpace.
Accessed 832 times total.

Classification:

54F99 (General topology :: Special properties :: Miscellaneous)

Pending Errata and Addenda

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