Bolzano Weierstrass Theorem for Nets?

Question

For real sequences, we know that:

(1) Every bounded sequence has a convergent subsequent.

(2) Every sequence has a monotone subsequence.

For nets, do we have the corresponding theorems? (Extra: Can you give me reference?) Thank you.

Answer 1

Theorem (1) is true for nets: any bounded net in $\mathbb{R}^n$ has a convergent subnet. Indeed, more generally, a topological space $K$ is compact iff every net in $K$ has a convergent subnet. This is a standard theorem you should be able to find in any text that covers nets. In fact, you can even find a proof on Wikipedia.

Theorem (2) is also true for nets: any net in $\mathbb{R}$ has a monotone subnet. To prove this, let $(x_i)$ be a net in $\mathbb{R}$. Passing to a subnet using Theorem (1), we may assume $(x_i)$ converges to some $x\in\mathbb{R}\cup\{-\infty,\infty\}$. Further passing to a subnet, we may assume either $x_i<x$ for all $i$, $x_i=x$ for all $i$, or $x_i> x$ for all $i$ (since at least one of these properties must hold for a cofinal set of $i$). The second case is trivial and the first and third cases are essentially equivalent; without loss of generality, suppose we are in the first case.

Let $(I,\leq)$ be the index set of the net and define a new order $\preceq$ on $I$ by saying $i\preceq j$ iff $i\leq j$ and $x_i\leq x_j$. I claim the order $\preceq$ is directed. Indeed, given $i,j\in I$, there exists $k$ such that $i\leq k$ and $j\leq k$. Since $x_i<x$ and $x_j<x$ and the net converges to $x$, there then exists $\ell\geq k$ such that $x_\ell>\max(x_i,x_j)$. We then have $i\preceq\ell$ and $j\preceq\ell$.

Thus $(I,\preceq)$ is also a directed set. The identity map $(I,\preceq)\to(I,\leq)$ is cofinal, and so we get a subnet of $(x_i)$ where the index set is ordered by $\preceq$ instead of $\leq$. This subnet is monotone increasing, by the definition of $\preceq$.

(This argument works with $\mathbb{R}$ replaced by any totally ordered set with the order topology, except that to find a limit $x$ of a subnet you may need to pass to the Dedekind completion of the totally ordered set.)