Lebesgue covering theorem

Lebesgue's covering theorem states that for any shape with a finite size in a space with dimension greater than or equal to two (like a plane), you can cover it completely with a collection of small, overlapping shapes (like circles or squares). Importantly, the overlaps are controlled so that the largest size of these shapes can be made arbitrarily small, ensuring precise coverage. This theorem guarantees that complex shapes can be effectively covered by simple, small elements, which is fundamental in mathematical analysis for understanding and approximating more complicated structures.