Definition
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[06P] Given a directed set \((X,β€_ X)\) a subset of it \(Yβ X\) is called cofinal if
\begin{equation} \label{eq:cofinale} β xβ X ~ β yβ Y,~ yβ₯_ X x \end{equation}
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More in general, another directed set \((Z,β€_ Z)\) is said to be cofinal in \(X\) if there exists a map \(i : Zβ X\) monotonic weakly increasing and such that \(i(Z)\) is is cofinal in \(X\); i.e.
\begin{equation} \label{eq:cofinale_ Z,X} ( β z_ 1,z_ 2β Z, z_ 1β€_ Z z_ 2β i(z_ 1)β€_ X i(z_ 2) ) ~ ~ β§~ ~ (β xβ X ~ β zβ Z,~ i(z)β₯_ X x) \end{equation}
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(This second case generalizes the first one, where we may choose \(i:Yβ X\) to be the injection map, and \(β€_ Y\) to be the restriction of \(β€_ X\) to \(Y\).)