Every Net on a Total Ordering has a Monotone Subnet

Last updated Nov 6, 2022

Let $(X, \leq)$ be a Total Ordering and let $(x_{\alpha}){\alpha \in I} \subset X$ be a Net on $X$, with Index Set $I$. Then there exists a Subnet $(y{\beta})_{\beta \in J} \subset X$ that is a Monotone Net.

Consider ${x_{\alpha}}_{\alpha \in I}$. If it has no Upper Bound, … TODO I think this needs Transfinite Induction

• Transfinite Induction

See Every Sequence on the Reals has a Monotone Subsequence for a template.