Metrizability of Mahavier Products Indexed by Partial Orders
Keywords:
compactness, inverse limits, LOTS, Mahavier products, metrizability, partial ordersAbstract
Let $X$ be separable metrizable, and let $f \subseteq X^2$ be a non-trivial relation on $X$. For a given partial order $\langle P, \leq\rangle$, the Mahavier product $\mathbf{M}\langle X, f, P\rangle \subseteq X^P$ (also known as a generalized inverse limit) collects functions such that $x(p) \in f(x(q))$ for all $p \leq$ q. Steven Clontz and Scott Varagona [Topology Proc. 54 (2019), 259-269] previously showed that for well orders $P, \mathbf{M}\langle X, f, P\rangle$ is separable metrizable exactly when $P$ is countable and $f$ satisfies condition $\Gamma$; we extend this result to hold for all partial orders.
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