Stability of persistence diagram on quotient spaces. The stability of persistence diagram, which provides validity of persistent homology, has been proved by Edesbrunner . It says that for persistence diagram, which is a multiset of points in the extended plane induced by a real-value function, that small changes in the filtration functions imply only small changes in the diagram: where f, g are two functions, ∥ • ∥ ∞ is the L ∞ norm, and d B is the bottleneck distance between two diagrams. From Lemma 4.1.1 in Proposition 4.1, it is always able to induce a filtration from group actions by isometries in . Let f ′ , g ′ be the filtration functions induced on orbit space by discrete group actions,then their induced bottleneck distance d B (D(f ′ ), D(g ′ )) = inf γ (sup x ′ ∈D(f ′ ) ∥x ′ -γ(x ′ )∥ ∞ ) = d B (D(f ), D(g)) = inf γ (sup x∈D(f ) ∥x -γ(x)∥ ∞ ) ≤ ∥f -g∥ ∞ , since the points on persistence diagrams of covering and underlying spaces always coincide when |G| is coprime to characteristic of field F (Theorem 5.1). Here, x ′ ∈ D(f ′ ),x ∈ D(f ) range over all points on two diagrams, and γ ranges over all bijections from D(f ) → D(g) and D(f ′ ) → D(g ′ ). Thus, we have shown that the induced persistence diagrams on orbit spaces also provide a stable representation if the original diagrams on covering spaces is stable.