= Solution
A <connected component> $U$ of $\rho_A(x)$ meets the relatively clopen subset $\rho_B(x)$ either everywhere or nowhere: its intersection and its complement would otherwise disconnect $U$. If it meets nowhere, $U\subseteq\sigma_B(x)$. If it meets everywhere, $U$ is a <connected> subset of $\rho_B(x)$, and any larger <connected> subset of $\rho_B(x)$ would be a <connected> subset of $\rho_A(x)$ larger than its component $U$. Thus \b[$U$ is either a whole component of $\rho_B(x)$ or lies wholly in $\sigma_B(x)$].
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