Finite character, monotonic functions, partial functions 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. 
http://en.wikipedia.org/wiki/Finite_character
 A: 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.
