A definition of directed set
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of set.
- The reader knows a definition of relation.
Target Context
- The reader will have a definition of directed set.
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.
Main Body
1: Definition
Any set, \(S\), with any relation on \(S\), \(\leq\), such that 1) \(p \leq p\) for every \(p \in S\); 2) if \(p_1 \leq p_2\) and \(p_2 \leq p_3\), \(p_1 \leq p_3\); 3) for every pair, \(p_1, p_2 \in S\), there is a \(p_3 \in S\) such that \(p_1 \leq p_3\) and \(p_2 \leq p_3\).
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 \(U_\alpha \leq U_\beta\) 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.
