Aronszajn tree

A set-theoretic tree of height ${\omega }_1$ with countable levels and no cofinal branch. More generally, a $\kappa$-Aronszajn tree has height a regular uncountable $\kappa$, levels of size less than $\kappa$, and no cofinal branch.

Given coherent coinfinite injections into omega, use all restrictions $e_{\alpha }\upharpoonright \beta$, where $\beta \le \alpha <{\omega }_1$, ordered by proper extension. Every level is countable because its nodes differ finitely from a fixed function on a countable domain. An uncountable chain in a partial order would have unbounded domain heights, and its union would inject ${\omega }_1$ into $\omega$, which is impossible.

An Aronszajn tree with no uncountable tree antichain. A normal splitting Suslin tree yields a Suslin line by a lexicographic ordering followed by Dedekind completion.

A well-pruned set-theoretic tree that is an Aronszajn tree and a Suslin tree gives a forcing with the countable chain condition for forcing: stronger nodes extend weaker ones. The ${\omega }_1$ dense subsets of a forcing order of nodes at or above each level cannot all be met by a filter in an ordered set, since that would produce a cofinal branch. Thus ${\mathrm{MA}}_{{\aleph }_1}$ fails. When the continuum exceeds ${\aleph }_1$, full Martin axiom includes this instance.

An Aronszajn tree that is a union of countably many tree antichains. Equivalently, it admits a map into a countable set that is injective on each chain.

A set-theoretic tree of height ${\omega }_2$, levels of size at most ${\aleph }_1$, and no cofinal branch. The Continuum hypothesis supplies one through the minimal-walk tree.