is the variable of time for filtration . We assume that all subspaces generated from filtration remain G-invariant, which is to say that the given filtration functional F is also invariant under group action. The complex K and subcomplex series are determined by function F and space X, thus one can say that F(X) coincides with F(K) hereafter. Embedded in a topological space X, complex K may be finite or infinite, leading to finite or infinite dimensional linear spaces of chain complexes respectively. However, considering the particles in a system is finite or countable, since the transformation group Γ is always discrete and not dense, we assume that these chain complexes are always countable. Hence, from basic functional theory, one can always find a generalized canonical basis set, consisting countable simplexes or cells as {σ i } spanning the space. In this article, we always assume that any filtration is G-invariant, thus the subspaces will share the same global symmetry condition with their original space. Definition 3.1 (Normal covering). A covering map induced from group actions is called normal if the covering G-space has the following property: Let p : X → X 0 , there has ∀x ∈ X 0 , then each pair of its fiber x, x′ , there is a deck transformation taking x → x′ .