Left: blue excursion set on the sphere consisting of an upper-left component with a hole, an upper-right component, and a lower component. Its Betti numbers are β₀ = 3, β₁ = 1, β₂ = 0, and its Euler characteristic is EC =3−1 + 0 = 2. Middle: pink mask in which the data are not reliable. The mask covers part of the upper-left component and hole; its hole is fully contained in the upper-right component, and it overlaps the lower component in two disconnected pieces. Right: visualization of the relative homology groups obtained by shrinking the mask to a point and pulling the excursion set with it. We have b₀ = 0 because all three components connect to the shrunken mask, b₁ = 2 because the loop in the upper-left component is preserved and a new loop in the lower component is formed, and b₂ = 1 because the upper-right component takes on the shape of a sphere. The (relative) Euler characteristic is therefore EC_rel = 0−2 + 1 = −1.