153: Directed Set

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A definition of directed set

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Topics

About: set

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Definition
• 2: Note

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Starting Context

• The reader knows a definition of set.
• The reader knows a definition of relation.

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Target Context

• The reader will have a definition of directed set.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Definition

Any set, $S$, with any relation on $S$, $\le$, such that 1) $p\le p$ for every $p\in S$; 2) if $p_1\le p_2$ and $p_2\le p_3$, $p_1\le p_3$; 3) for every pair, $p_1,p_2\in S$, there is a $p_3\in S$ such that $p_1\le p_3$ and $p_2\le p_3$.

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2: Note

The relation can be partial: a pair, $p_1,p_2\in S$, may not be related.

As an example, the topology (which is the set of all the open sets) on a Euclidean topological space with the relation that $U_{\alpha }\le U_{\beta }$ if and only if $U_{\beta }\subseteq U_{\alpha }$ is a directed set, but the relation is partial. It is a directed set because 1) $U_{\alpha }\subseteq U_{\alpha }$; 2) if $U_{\beta }\subseteq U_{\alpha }$ and $U_{\gamma }\subseteq U_{\beta }$, $U_{\gamma }\subseteq U_{\alpha }$; 3) $U_{\alpha }\cap U_{\beta }\subseteq U_{\alpha }$ and $U_{\alpha }\cap U_{\beta }\subseteq U_{\beta }$. But the relation is partial because there are 2 open sets such that neither $U_{\beta }\subseteq U_{\alpha }$ nor $U_{\alpha }\subseteq U_{\beta }$, for example 2 disjoint open sets.

The natural numbers set (whether it includes $0$ or not) with the canonical relation is a directed set.

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References

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