Given a finite Weyl group W_fin with root system Phi_fin, one can create the affine Weyl group W_aff by taking the semidirect product of the translation group associated to the coroot lattice for Phi_fin, with W_fin. The double affine Weyl semigroup W can be created by using a similar semidirect product where one replaces W_fin with W_aff and the coroot lattice with the Tits cone of W_aff. We classify cocovers and covers of a given element of W with respect to the Bruhat order, specifically when W is associated to a finite root system that is irreducible and simply laced. We show two approaches: one extending the work of Lam and Shimozono, and its strengthening by Milicevic, where cocovers are characterized in the affine case using the quantum Bruhat graph of W_fin, and another, which takes a more geometrical approach by using the length difference set defined by Muthiah and Orr. / Doctor of Philosophy / The Bruhat order is a way of organizing elements of the double affine Weyl semigroup so that we have a better understanding of how the elements interact. In this dissertation, we study the Bruhat order, specifically looking for when two elements are separated by exactly one step in the order. We classify these elements and show that there are only finitely many of them.