We study the problem of the lower bounds of the modulus of families of paths of order p, \(p>n-1,\) and their connection with the geometry of domains containing the specified families. We have proved an analogue of Näkki’s theorem on the positivity of the p-modulus of families of paths joining a pair of continua in the given domain. Besides that, we show that domains with a p-strongly accessible boundary with respect to p-modulus, \(p>n-1,\) are finitely connected at their boundary. The mentioned result generalizes Näkki’s theorem, which was proved for uniform domains in the case of a conformal modulus.
