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Conditions with the same finite stem in Hechler forcing are compatible: $(s,f)$ and $(s,g)$ have the common stronger condition $(s,\max \{f,g\})$. Since there are only countably many finite stems, Hechler forcing is sigma-centered and hence has the countable chain condition for forcing. It therefore preserves ${\aleph }_1^M$.

In $M$, the Continuum hypothesis givesEvery real in $M[G]$ has a nice name for a real, and the countable chain condition bounds the number of such names bywhere the last equality uses the ground-model continuum hypothesis. The extension still contains all ground-model reals, already ${\aleph }_1$ many, soThus forcing once with $\mathbb{D}$ preserves the continuum hypothesis.

A forcing order is sigma-centered when it is the union of countably many centered subsets. Every sigma-centered forcing has the countable chain condition for forcing.

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.