This paper is devoted to the study of epsilon-entropy of the Nikolsky classes H(alpha, infinity)(I(s)) in C(K), where K is an arbitrary compact set in I(s). For a connected set K the order of epsilon-entropy is known to be the same as the order of Komolgorov's epsilon-dimension. Without connectedness this is not the case. The exact order is given in terms of two functions characterizing "density" and "discontinuty" of the compact K.