Do nets have subsequence?

Question

Let $(P,\leq)$ be a directed set. Is there a cofinal and increasing function $\theta:\mathbb{N}\to P$. if $P$ has any maximal elements it can be proved easily. So I suppose $(P,\leq)$ none of it's elements is maximal. ($\mathbb{N}=\mathbb{Z}^+$)

Answer 1

Not necessarily. For example, $P$ could be $\omega_1$, the set of countable ordinals, with the usual ordering: no countable subset of $\omega_1$ is cofinal. Or let $X$ be any uncountable set, and let $P$ be the set of finite subsets of $X$: $\langle P,\subseteq\rangle$ is a directed set, but no $\theta:\Bbb Z^+\to P$ is cofinal. To see this, merely note that if $\theta:\Bbb Z^+\to P$ is any function, $\bigcup_{n\in\Bbb Z^+}\theta(n)$ is a countable subset of $X$, so for any $x\in X\setminus\bigcup_{n\in\Bbb Z^+}\theta(n)$, $\{x\}\nsubseteq\theta(n)$ for all $n\in\Bbb Z^+$, and $\theta$ is therefore not cofinal in $P$.