During the past decade, the landscape of condensed matter physics has been dominated by a desire to understand the topological nature of materials and the observable consequences that are associated with the presence of topology. To this end, topological band theory has provided a foundation that has allowed for the definition of topological invariants, or conserved quantities that do not change based on adiabatic changes to the system parameters. Nonetheless, each of the systems that has been considered to this point, have relied on the fact that the system under consideration is both closed and Hermitian. Recent work has extended topological band theory to open, non-Hermitian Hamiltonians but little is understood about how non-Hermicity alters the topological quantization of associated observables. In this talk, we will begin to address these problems by examining the non-Hermitian Chern insulator where we focus on changes in observables and its relation to our current understanding about non-Hermitian topological band theory.