Topological features born and dying as scale parameter ε grows
VR(ε) = {σ ⊆ X : d(x,y) ≤ ε ∀ x,y ∈ σ} — clique complex at scale ε H₀ (β₀): connected components — born at 0, die when merged H₁ (β₁): loops/holes — born when loop closes, die when filled
Stability: d_bottleneck(Dgm(f),Dgm(g)) ≤ ‖f−g‖∞ (Cohen-Steiner et al 2007)