Before revision:  We provide stability results for the cover filtration and show how the construction is equivalent to the Cech filtration under certain settings  .
After revision:  We prove a stability result: the cover filtrations of two covers are \alpha/m interleaved, where \alpha is a bound on bottleneck distance between covers and m is the size of smallest set in either cover. We also show our construction is equivalent to the Cech filtration under certain settings .