Two Metres Apart: A Rigorous Topological and Metric Framework for Physical Distancing Policies

The feasibility of physical-distancing interventions during the COVID-19 pandemic implicitly relied on structural properties of physical space. We show that minimum-distance policies (usually formulated as “individuals must remain at least δ metres apart”) require not only the Hausdorff property but a compatible metric structure that supports uniform separation with a positive lower bound. Using results from metric topology, we prove that the existence of disjoint open neighbourhoods is necessary but insufficient for well-posed quantitative distancing constraints. We demonstrate using quo-tient and identification topologies arising in epidemiological modelling (like grid aggregation, mean-field limits) that distancing can become ill-defined even when Hausdorffness is preserved. We discuss implications for spatial epidemiology, where metric and topological assumptions are typically implicit but essential for distance-dependent transmission kernels.