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With reference to Folland's Real Analysis on the following definition;

An increasing cofinal function $\varphi:B \rightarrow A$ from a directed system $B$ into a directed system $A$ is a mapping such that $\beta_{1} \leqslant \beta_{2}$ implies $\varphi(\beta_{1}) \leqslant \varphi(\beta_{2})$, and for every $\alpha \in A$ there is $\beta \in B$ such that $\alpha \leqslant \varphi(\beta)$.

I would like to know if the directed system imply here have the same meaning as directed set?

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A directed system is a set $S$ with a relation $\le$ that satisfies $\forall s \in S: s \le s$ and $\forall s,t,u \in S: (s \le t \land t \le u) \rightarrow s \le u$, so a reflexive and transitive relation (it need not a full partial order). I think most texts would define directed set in the same way, e.g. wikipedia does.

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Yes, it seems that it has the same meaning. I have also heard that directed system is a functor from directed set to some category and its colimit (which may not exist) is called direct limit.

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