\(DMC\): Dependent Multiple Choice: Every tree in which every element has at least one immediate successor has a subtree whose levels are finite and in which every element has at least one immediate successor.

Weak DC: If \(R\) is a binary relation on a non-empty set \(E\) such that \((\forall x\in E)(\exists y\in E)( x\mathrel R y)\), then there is a sequence \(\langle F_n\rangle_{n\in\omega}\) of non-empty finite subsets of \(E\) such that \((\forall n\in\omega) (\forall x\in F_n)(\exists y\in F_{n+1}) (x\mathrel R y)\). (This statement remains equivalent to Form 106 if we replace "binary relation" by "transitive binary relation".)

Every scattered compact space is a Baire space.

For every compact \(T_2\) space \((X,T)\) and every family \(D=\{\, D_i : i\in\omega\,\}\) of dense open sets of \(X\) there is a regular filter base \(\cal F \subseteq T\) (that is, if \(F,G\in \cal F\), then there exists \(Q\in \cal F\) such that \(\overline{Q}\subseteq F\cap G\)) such that for all \(i\in\omega\), for all but finitely many \(F\in\cal F\), \(\overline{F}\subset D_i\).

Baire Category Theorem for Locally Compact Regular Spaces: Every locally compact regular space is Baire.