While compactness in the current topology is elementary, the claim that the limit is also an integral current is nontrivial, and can be viewed as a regularity result. The same holds with the Allard compactness theorem below. For our applications, we will always work with closed integral currents, namely $\partial L_{i}=0$, which implies $\partial L=0$ in the limit. To such currents one can associate a homology class.