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Definition 119

[06P] Given a directed set $(X, \leq_{X})$ a subset of it $Y \subseteq X$ is called cofinal if

\begin{matrix} (1) & \forall x \in X \exists y \in Y, y \geq_{X} x \end{matrix}

120

More in general, another directed set $(Z, \leq_{Z})$ is said to be cofinal in $X$ if there exists a map $i : Z \to X$ monotonic weakly increasing and such that $i (Z)$ is is cofinal in $X$ ; i.e.

\begin{matrix} (2) & (\forall z_{1}, z_{2} \in Z, z_{1} \leq_{Z} z_{2} \Rightarrow i (z_{1}) \leq_{X} i (z_{2})) \land (\forall x \in X \exists z \in Z, i (z) \geq_{X} x) \end{matrix}

121

(This second case generalizes the first one, where we may choose $i : Y \to X$ to be the injection map, and $\leq_{Y}$ to be the restriction of $\leq_{X}$ to $Y$ .)

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Authors: "Mennucci , Andrea C. G." .

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cofinal
order, with filtering property
order, directed
order, with filtering property
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