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Both Cantor middle-third set and Sierpi\'nski carpet are self-similar,perfect, compact metric spaces. In spite of the similarity of the mathematicalprocedure of construction, there exists between them a fundamental differencein topological nature, and this difference affects the methods of constructionof an interesting non-trivial quotient space of them. The totallydisconnectedness (or, more generally, zero-dimensional) enables Cantormiddle-third set to have a non-trivial quotient space which is self-similar. Onthe other hand, concerning Sierpi\'nski carpet, because of the connectedness ofits structure, no non-trivial quotient space which is self-similar can beconstructed by such an elegant procedure as that for Cantor middle-third set.Various topologically significant nature specific to Cantor middle-third setowe mainly to the totally disconnectedness of the set.

On a difference in topological nature between Cantor middle-third set and Sierpi\'nski carpet