In recent years, non-Hermitian physics has revealed numerous novel phenomena in open quantum systems, such as the non-Hermitian skin effect, which challenges the conventional bulk-boundary correspondence. This paper proposes a generalized index theorem that unifies the description of topological characteristics in nonHermitian systems by introducing biorthogonal Fredholm modules and a skin effect correction term. The core formula is the skin indicator, which combines the topological invariants of the classical Atiyah-Singer index theorem with a Jump term that quantifies the breakdown of the bulk-boundary relationship. This framework is applied to chiral edge currents in the quantum Hall effect and impedance quantization in non-Hermitian topological circuits, revealing the intrinsic mechanisms of non-reciprocal topological phase transitions. The work presented here provides new theoretical tools for the topological analysis of open systems