We study the reverse mathematics of characterization theorems of regular countable second countable (CSC) spaces. We prove that arithmetic comprehension is equivalent over $\mathbf {RCA}_0$ to every $T_3$ CSC space being metrizable, and we characterize the $T_3$ spaces which are metrizable over $\mathbf {RCA}_0$. We show that Lynn’s theorem for CSC spaces can be carried out in $\mathbf {ACA}_0$, 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 $T_2$ compact CSC space being well-orderable. From general topology, we know that the locally compact $T_2$ CSC spaces are the well-orderable CSC spaces, and that the $T_3$ scattered CSC spaces are the completely metrizable CSC spaces. We show that these characterizations and a few others are equivalent to arithmetic transfinite recursion over $\mathbf {RCA}_0$. We also find a few statements that are equivalent to $\Pi ^1_1$ comprehension. In particular we show that every $T_3$ CSC space has a Cantor–Bendixson rank and that every $T_3$ CSC space is the disjoint union of a scattered space and dense in itself space are equivalent to $\Pi ^1_1$ comprehension over $\mathbf {RCA}_0$.