The notion of the strong connectness of Sobolev spaces was determined by E. Ya. Khruslov [Mat. Sb., N. Ser. 106(148), 604-621 (1978; Zbl 0381.35032)]. This property plays an important role in the study of the convergence of variational problems solutions in perforated domains. In this paper the authors establish the conditions on a weighted function, which are ensured a certain compactness of the sequences of functions with different domains of definitions and the strong connectness of corresponding Sobolev spaces for some type of perforated domains.