Compact batching of a small open set

An open subset of the circle can be covered by countably many closed subarcs with disjoint interiors. Grouping a fixed enumeration into finite batches gives compact sets covering every point of the open set. By taking each finite batch long enough, the remaining total measure can be forced below any prescribed positive sequence tending to zero. This controls the measure of subsequent batches without assuming that a prescribed null subset is itself a countable union of compact null sets.