Difference in topological characteristics between cantor middle-third set and Sierpiński carpet

A difference in topological nature between Cantor middle-third set and Sierpiński carpet, both of which are perfect, compact and self-similar spaces has been made clear through the investigations of the procedure of construction of an interesting quotient space. In spite of the similarity of the mathematical procedure of construction, there exists a fundamental difference in topological nature between them, and this difference affects the methods of construction of an interesting non-trivial quotient space of them. The totally disconnected nature (or, more generally, zero-dimensional) enables Cantor middle-third set to have a non-trivial quotient space which is self-similar. On the other hand, concerning Sierpiński carpet, because of the connectedness of its structure, no non-trivial quotient space which is self-similar can be constructed by such an elegant procedure as that for Cantor middle-third set. Various topologically significant nature specific to Cantor middle-third set owe mainly to the totally disconnected nature of the set.