Consequently, it is suggested a quotient filtered space and its Γ-invariant covering space would coincide in their bar-code diagrams if coefficients in F has a charateristic either zero or comprime to |Γ|. As the persistent groups in K or K/Γ will have to share a equal birth time (index) and death time (index), where H l,p k (K) = 0 ⇔ H l,p k (K/Γ) = 0. This validates the consideration of calculating persistent homology in a quotient space induced by group actions, rather than its covering, while still keeping principal information of topological variables invariant. As a complementary, we consider some infinite but discrete groups of Γ = Z n , where the n-dim lattice group acting on R n spaces. Corollary 5.1.1. Let X = K = R n and Γ = Z n , then: