Could you explain to me why family of partial functions and family of monotonic functions are of finite character?
I'm asking this because I'm currently reading a proof of a theorem concerning a bijection between two well-ordered sets, and it's using Zorn's lemma. We pick a chain in a partially ordered (by inclusion) set of partial functions and we take its union as its upper bound. And then it's written that because both being a partial function and being a monotonic function have finite character and the chain satisfies these conditions, then its union also satisfies them.
Thank you.
PS. I've tried looking for it in the internet, but I didn't find much. Just a few lines about Teichmuller-Tukey Lemma on Wikipedia, etc.
Let $\mathscr{F}$ be the family of partial functions from $X$ to $Y$. Suppose first that $f\in\mathscr{F}$; then certainly every finite subset of $f$ is a partial function from $X$ to $Y$. Suppose, on the other hand, that every finite subset of some set $f$ belongs to $\mathscr{F}$. Clearly $f\subseteq X\times Y$, so to see that $f\in\mathscr{F}$ we need only show that if $\langle x,y_0\rangle,\langle x,y_1\rangle\in f$, then $y_0=y_1$. If $\langle x,y_0\rangle,\langle x,y_1\rangle\in f$, then $\big\{\langle x,y_0\rangle,\langle x,y_1\rangle\big\}$ is a finite subset of $f$, so $\big\{\langle x,y_0\rangle,\langle x,y_1\rangle\big\}\in\mathscr{F}$; i.e., $\big\{\langle x,y_0\rangle,\langle x,y_1\rangle\big\}$ is a partial function from $X$ to $Y$, and therefore $y_0=y_1$ as desired. Thus, $\mathscr{F}$ has finite character.
The argument for monotone increasing is very similar. If every finite subset of a partial function $f$ from $X$ to $Y$ is monotone increasing, then in particular every two-element subset of $f$ is increasing. Thus, for any two points $x_0,x_1$ in the domain of $f$ with $x_0<x_1$, the restriction $f\upharpoonright\{x_0,x_1\}$ is increasing, and $f(x_0)<f(x_1)$. But this means that $f$ itself is increasing on its domain.