Abstract: By embedding the dynamic characteristics into the finite Cantor set and improving it, P-sets were proposed. P-sets have dynamic characteristics. P-sets are a pair of sets composed of internal P-set XI'(internal packet set XI')and outer p-set XF(outer packet set XF).By using P-sets, the concepts of internal P-equivalence classes, outer P-equivalence classes and P-equivalence classes were presented. The theorems were obtained such as the P-equivalence classes reduction theorem, the internal point theorem of internal P-equivalence classes discrete interval, the external point theorem of outer P-equivalence classes discrete interval,the subinterval theorem of internal P-equivalence classes discrete interval, and P-equivalence classes identification criterion. Using these results, the applications of P-equivalence classes in searching-identification about unknown information were given. The paper shows that there are overlapping and osmosis spaces between P-sets and cantor sets, and some new results are hidden in them.

Key words: P-sets, P-equivalence classes, Internal point theorem, External point theorem, Subinterval theorem, Applicataon