In this paper, we introduce the notion of topological UP-algebras and several types of subsets of topological UP-algebras, and prove the generalization of these subsets. We also introduce the notions of quotient topological spaces of topological UP-algebras and topological UP-homomorphisms. Furthermore, we study the relation between topological UP-algebras, Hausdor spaces, discrete spaces, and quotient topological spaces, and prove some properties of topological UP-algebras.