Metrizability of Mahavier products indexed by partial orders

TL;DR

This paper extends the characterization of when Mahavier products indexed by partial orders are separable metrizable, generalizing previous results from well orders to all partial orders.

Contribution

It generalizes the conditions under which Mahavier products are separable metrizable from well orders to arbitrary partial orders.

Findings

Mahavier products are separable metrizable if and only if the index partial order is countable.

The condition on the relation $f$ (condition $ ext{ extGamma}$) is necessary for metrizability.

The result broadens the understanding of inverse limits indexed by complex order structures.

Abstract

Let $X$ be separable metrizable, and let $f\subseteq X^2$ be a non-trivial relation on $X$. For a given partial order $(P,\leq)$, the Mahavier product $M(X,f,P)\subseteq X^P$ (also known as a generalized inverse limit) collects functions such that $x(p)\in f(x(q))$ for all $p\leq q$. Clontz and Varagona previously showed for well orders $P$ that $M(X,f,P)$ is separable metrizable exactly when $P$ is countable and $f$ satisfies condition $\Gamma$; we extend this result to hold for all partial orders.