Metrization of the space of all simple rings in the region of family $\mathfrak B$

The problem involved with metrizing space $\Sigma$ is solved. In particular, it is shown that simple rings in the sequence of plane regions $(B_n)$ converging to kernel $B_0$ can be introduced by properly metrizing the set $\bigcup\limits_{n=0}^\infty{B_n}$.