Local reflections of choice

Under the assumption of small violations of choice with seed \(S\) \((SVC(S))\), the failure of many choice principles reflect to local properties of \(S\), which can be a helpful characterisation for preservation proofs. We demonstrate the reflections of \(DC\), \(AC_\lambda\), \(PP\), and other important forms of choice. As a consequence, we show that if \(S\) is infinite then \(S\) can be partitioned into \(\omega\) many non-empty subsets.