A more technical version of Federer-Fleming compactness replaces the current topology by the flat norm topology, which is a slightly stronger topology. The flat norm of an integral current $T$ is

$$\left\lVert T\right\rVert_{flat}=\inf\{\text{Mass}(A)+\text{Mass}(B)|T=A+\partial B,\quad A,B\text{ are integral currents}\},$$

and the convergence $T_{i}\to T$ in this topology simply means $\left\lVert T-T_{i}\right\rVert_{flat}\to 0$.