REVERSE MATHEMATICS OF REGULAR COUNTABLE SECOND COUNTABLE SPACES

Abstract We study the reverse mathematics of characterization theorems of regular countable second countable (CSC) spaces. We prove that arithmetic comprehension is equivalent over RCA 0 $\mathbf {RCA}_0$ bold upper R upper C upper A 0 to every T 3 $T_3$ upper T 3 CSC space being metrizable, and we characterize the T 3 $T_3$ upper T 3 spaces which are metrizable over RCA 0 $\mathbf {RCA}_0$ bold upper R upper C upper A 0 . We show that Lynn’s theorem for CSC spaces can be carried out in ACA 0 $\mathbf {ACA}_0$ bold upper A upper C upper A 0 , namely that every zero-dimensional separable space is homeomorphic to the order topology of a linear order. We also show that arithmetic comprehension is equivalent to every T 2 $T_2$ upper T 2 compact CSC space being well-orderable. From general topology, we know that the locally compact T 2 $T_2$ upper T 2 CSC spaces are the well-orderable CSC spaces, and that the T 3 $T_3$