is homeomorphic to the product space A × I where I has the discrete topology. Preservation of topological properties Every disjoint union of discrete spaces is discrete Separation Every disjoint union of T0 spaces is T0 Every disjoint union of T1 spaces is T1 Every disjoint union of Hausdorff spaces is Hausdorff Connectedness The disjoint union of two or more nonempty topological spaces is disconnected See also product topology, the dual construction subspace topology and its dual quotient topology topological union, a generalization to the case where the pieces are not disjoint General topology