In this paper, we show how to associate certain topologies with special ideals of BCC-algebras on these BCC-algebras. We show that it is natural for BCC-algebras to be topological BCC-algebras with respect to theses topologies. Furthermore, we show how certain standard properties may arise. In addition we demonstrate that it is natural for these topologies to have many clopen sets and thus to be highly connected via the ideal theory of BCC-algebras.