Reverse mathematics of separably closed sets (Q2580957)

This short note is a contribution to the program of reverse mathematics (\textit{S. G. Simpson}'s book [Subsystems of second order arithmetic. Springer, Berlin (1999; Zbl 0909.03048)] is the main reference). \textit{D. K. Brown} [in: Logic and computation, Proc. Workshop, Pittsburgh/PA (USA) 1987, Contemp. Math. 106, 39--50 (1990; Zbl 0693.03041)] studied different notions of closed set in Polish spaces and the strength of statements about the equivalence of these notions. In particular he claimed the equivalence of the statement ``in a compact Polish space every closed set is separably closed''\ to \(\mathbf{ACA}_0\) over the weak base system \(\mathbf{RCA}_0\). Brown's proof of the reversal (i.e.\ the fact that \(\mathbf{RCA}_0\) plus the statement imply \(\mathbf{ACA}_0\)) however had a problem, and in this note Hirst provides a correct proof of this implication.