\(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\)  is  a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\)  then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\).  See Tarski [1948], p 96, Levy [1964], p. 136.

Martin's Axiom \((\aleph_{0})\): Whenever \((P\le)\) is a non-empty, ccc  quasi-order (ccc means every anti-chain is countable) and \({\cal D}\) is a family of \(\le\aleph_0\) dense subsets of \(P\), then there is a \({\cal D}\) generic filter \(G\) in \(P\).