Notes on quasi-metrizability and trees
Keywords:
quasi-metric, tree, chain, antichain, branch, stationary, non-Archimedean, full binary tree of height $\omega$, $L$-labeling, $L$-special, forcing, absoluteAbstract
Without trying to be comprehensive, this article introduces some necessary and sufficient conditions under which a tree is quasi-metrizable. Sufficient conditions include the concepts of $\mathbf{Q}$-special and $\mathbb{R}$-special, and necessary ones include height $\leq \omega_1$ and lack of uncountable branches. It is shown how a Souslin tree is not quasi-metrizable by using a forcing and absoluteness argument, and the same is done for a tree defined in ZFC.
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