where k 1 < k 2 < ... < k n represent our filtration thresholds, and M represents the original manifold. We can measure/quantify the topology of these nested sublevel sets with the EC at each filtration threshold {χ 1 , χ 2 , ..., χ n }. We ultimately obtain an ordered pair of values {k i , χ i }, which characterize the topology of the manifold and its associated function.